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On some non-Archimedean normed linear spaces Robert, Joseph Pierre 1965

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The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of J, PIERRE ROBERT FRIDAY, APRIL 2, 1965, at 3:30 P.M. IN ROOM 102, ARTS BUILDING COMMITTEE I.N CHARGE Chairman: I. MoT. Cowan Do Jo Bures C..W. Clark No J„ Divinsky Maurice Sion Avrom Soudack C, A. Swanson External Examiner:. Harry F. Davis •University of Waterloo Waterloo, Ontario ON SOME NON-ARCHIMEDEAN NORMED LINEAR SPACES ABSTRACT A class of complete non-Archimedean pseudo-normed l i n e a r spaces for which the f i e l d of scalars has a t r i v i a l valuation i s introduced; we c a l l these spaces "V-spaces." V-spaces d i f f e r from the c l a s s i c a l normed l i n e a r spaces i n that the homogeneity of the norm, i s replaced by the requirement that He* x f| =: ||x|| for a l l x and a l l scalars c* f 0; the usual t r i a n g l e i n e q u a l i t y i s modified to for a l l x, y i f ||x|| i ||y|| , and i t i s assumed that the norm of an element i s either zero or i s equal to ^>n for a fixed realms "p* 1 and some integer n. The concept of a "distinguished basis" i n a V-space i s defined, By use of' a modified form of Riesz's Lemma,, i t . i s shown that every V-space admits a distinguished basis. Each element of a V-space then has a uniquely determined series expansion i n terms of the elements of a given distinguished bases. An analogue of the Paley-Wiener Theorem, i s proved f o r distinguished bases. Properties of distinguished bases are exploited throughout t h i s work. Linear and non-linear operators on V-spaces are also studied. In the usual way, a norm i s defined under which the set of bounded operators i s a V-space and the set of bounded l i n e a r operators i s a "V-algebra." A ch a r a c t e r i z a t i o n of bounded l i n e a r operators i s given as well as theorems on spectral decompositions. Under c e r t a i n assumptions on the expansions of x, y, A, the existence of solutions to equations of the form xz = y i n V-algebras, and of the form Ax = y i n ar b i t r a r y V-spaces i s proved. Approximations of the solutions are obtained. ||x + y|| f<Max (||x|| , ||y||} t "Max {||x|| , ||y||j A representation theorem f o r continuous l i n e a r functionals on a V^space i s given. This representation uses an analogue of the c l a s s i c a l inner, product. Examples of V"°spaces and V-algebras discussed include spaces of functions from a Hausdor.fi" space to a normed l i n e a r space 5 on which, the pseudo-norm characterizes the asymptotic behaviour of the functions. Some re s u l t s of the theory of pure asymptoti.es are extended to a r b i t r a r y V=spaces» GRADUATE STUDIES F i e l d of Study % Asymptotic Analysis i n Linear Spaces (Mathematics) Real Analysis Abstract Algebra Topics i n Algebra Point Set Topology Functional Analysis Asymptotic Expansions Theory of Functions M. Benedicty Bomshik Chang . de B„ Robinson So Cleveland C, Wo Clark Co A. Swanson R. A. Cleveland ON SOME NON-ARCHIMEDEAN NORMED LINEAR SPACES by J 0 / PIERRE ROBERT A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of MATHEMATICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF March BRITISH 1965 COLUMBIA In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t . c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission® T l £ K R £ 7£ Q B E R T Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 S Canada In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study* I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s - I t ' i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s for f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission... Department of '7, 0 _ _ R " 7 ~ The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date Afr.i£ J<>tS Abst r a c t A c l a s s of complete non-Archimedean pseudo-normed l i n e a r spaces f o r which the f i e l d of s c a l a r s has a t r i v i a l v a l u a t i o n i s i n t r o d u c e d ; we c a l l these spaces "V-spaces." V-spaces d i f f e r from the c l a s s i c a l normed l i n e a r spaces i n that the homogeneity of the norm i s r e p l a c e d by the requirement that |]cxx|f = |jx|| f o r a l l x and a l l s c a l a r s a £ 0; the us u a l t r i a n g l e i n e q u a l i t y i s m o d i f i e d to < Max [|]x||, ||y|1 } f o r a l l x, y, = Max (||x||, llyll} i f || x|| f llyll, x + y and i t i s assumed that the norm of an element i s e i t h e r zero or i s equal to p 1 1 f o r a f i x e d r e a l p > 1 and some i n t e g e r n. The concept of a " d i s t i n g u i s h e d b a s i s " i n a V-space i s d e f i n e d . By use of a m o d i f i e d form of Rie s z ' s Lemma, i t i s shown that every V-space admits a d i s t i n g u i s h e d b a s i s . Each element of a V-space then has a uniquely determined s e r i e s expansion i n terms of the elements of a given d i s t i n g u i s h e d b a s i s . An analogue of the Paley-Wiener Theorem i s proved f o r d i s t i n g u i s h e d bases. P r o p e r t i e s of d i s t i n g u i s h e d bases are e x p l o i t e d throughout t h i s work. Line a r and n o n - l i n e a r operators on V-spaces are al s o s t u d i e d . In the usual way, a norm i s d e f i n e d under which the set of bounded operators i s a V-space and the set of bounded l i n e a r operators i s a '*V-algebra." A c h a r a c t e r i z a t i o n of bounded l i n e a r operators i s given as. w e l l as theorems on i i i s p e c t r a l decompositions. Under c e r t a i n assumptions on the expansions of x, y, A, the e x i s t e n c e of s o l u t i o n s to equations of the form xz = y i n V-algebras, and of the form Ax = y i n a r b i t r a r y V-spaces i s proved. Approximations of the s o l u t i o n s are obtained. A r e p r e s e n t a t i o n theorem f o r continuous l i n e a r f u n c t i o n a l s on a V-space i s given. This r e p r e s e n t a t i o n uses an analogue of the c l a s s i c a l i n n e r product. Examples of V-spaces and V-algebras d i s c u s s e d i n c l u d e spaces of f u n c t i o n s from a Hausdorff space to a normed l i n e a r space, on which the pseudo-norm c h a r a c t e r i z e s the asymptotic behaviour of the f u n c t i o n s . Some r e s u l t s of the theory of pure asymptotics are extended to a r b i t r a r y V-spaces. i v Table of Contents Page INTRODUCTION 1 CHAPTER 1 Valued Spaces 1-1 D e f i n i t i o n s and n o t a t i o n s 6 1 -2 Some t o p o l o g i c a l p r o p e r t i e s 9 1-3 Some p r o p e r t i e s of the norm f u n c t i o n 13 1-4 Convergence of sequences and s e r i e s 16 1-5 Compactness 17 1-6 M o d i f i c a t i o n of Ri e s z ' s Lemma 1 9 1 - 7 D i s t i n g u i s h e d bases 22 CHAPTER 2 V-Spaces 2 - 1 D e f i n i t i o n s • 3 0 2 - 2 E x i s t e n c e of d i s t i n g u i s h e d bases 3 2 2 - 3 D i s t i n g u i s h e d f a m i l i e s of subsets 35 2 - 4 D i s t i n g u i s h e d complements 4 0 2 - 5 Notes 4 2 2 - 6 V-Algebras 44 CHAPTER 3 Examples of V-Spaces. Asymptotic and Moment Spaces. 3 - 1 I n t r o d u c t i o n 52 3 - 2 The 0 and o r e l a t i o n s 52 3~3 Asymptotic spaces: D e f i n i t i o n 54 3 - 4 Asymptotic spaces: Example I 61 3 - 5 Asymptotic spaces: Example II 65 V Page 3 - 6 Asymptotic spaces: Example I I I 6 7 3 - 7 Moment spaces 7 0 CHAPTER 4 s Bounded operators on V-spaces 4 - 1 D e f i n i t i o n s and n o t a t i o n s 7 8 4 - 2 The spaces ©*(Z, Y) and 0*(X) of bounded operators 80 4 - 3 The spaces 3(Z„ Y) and tJ (X) of bounded l i n e a r operators 8 3 4 - 4 C h a r a c t e r i z a t i o n of bounded l i n e a r operators 8 8 4 - 5 Inverses and s p e c t r a i n *3 (X) 9 4 4 - 6 Complete s p e c t r a l decompositions 9 5 4 - 7 Note on p r o j e c t i o n s 1 0 2 CHAPTER 5 : S o l u t i o n of equations 5 - 1 I n t r o d u c t i o n 1 0 5 5 - 2 Equations i n V-algebras 1 0 5 5 - 3 The equation Ax = y 1 1 1 5 - 4 The equation Ax = y i n v o l v i n g expansions of A and y 1 1 5 CHAPTER 6: Continuous l i n e a r f u n c t i o n a l s 6 - 1 Dual space 1 2 1 6 - 2 The *norm on (H) 1 2 3 6 - 3 H-inner product and r e p r e s e n t a t i o n theorems 1 2 5 B i b l i o g r a p h y I 3 2 Notation Index I 3 6 Subject Index 1 3 7 v i Acknowledgment I wish to express my g r a t i t u d e to P r o f e s s o r Charles A„ Swanson, whose n e v e r - f a i l i n g i n t e r e s t and advice have been i n v a l u a b l e to me=. I a l s o wish to thank P r o f e s s o r C o l i n W„ C l a r k f o r h i s c o n s t r u c t i v e c r i t i c i s m of the d r a f t form of t h i s work o The generous f i n a n c i a l support of l ' U n i v e r s i t e de Montreal and the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknows-.;.-. ledged. INTRODUCTION -The purpose of t h i s work i s to i n i t i a t e the theory of a non-standard type of pseudo-normed l i n e a r spaces, h e r e i n c a l l e d V-spaces. V-spaces depart from the c l a s s i c a l normed l i n e a r spaces ( [ 7 ] , [ 3 6 ] ) i n that the usual requirements on the norm f u n c t i o n ( 0 . 1 ) ||axj| = |a| f|xf| f o r a l l x and a l l s c a l a r s a, (0.2) ||x + y[| < ||x[| + [|y|| f o r a l l x, y, are r e p l a c e d by ( 0 . 3 . ) f|ax|) = ||x|| f o r a l l x and a l l s c a l a r s a t 0 , (0.4) II x + y| < Max (J|x||, f|y||] f o r a l l x, y, [ - Max {||*||. »y||} i f I N * llyll and, a l s o , by the a d d i t i o n a l c o n d i t i o n that the norm of an element i s e i t h e r 0 or i s equal to p n f o r a f i x e d r e a l p, 1 < p < 0 0 , and some i n t e g e r n. A V-space i s assumed to .be com-p l e t e with respect to i t s norm and the f i e l d of s c a l a r s to have c h a r a c t e r i s t i c 0 ( [ 1 0 ] ) . Thus, in the usual terminology, a V-space i s a complete s t r o n g l y non-Archimedean pseudo-normed l i n e a r •space over a f i e l d of s c a l a r s with c h a r a c t e r i s t i c 0 and a t r i v i a l v a l u a t i o n . The author's a t t e n t i o n was d i r e c t e d to t h i s a b s t r a c t s t r u c t u r e by the f o l l o w i n g example. A c l a s s i c a l method to obtain i n f o r m a t i o n about the asymptotic behaviour of a r e a l valued 2 f u n c t i o n i s to compare i t with the elements of an "asymptotic sequence" of f u n c t i o n s (see E r d e l y i [ 9 ] , van der Corput [ 3 ^ ] , [ 3 9 ] ) . C. A. Swanson and M. S c h u l z e r [ 3 2 ] , [ 3 3 ] , have extended t h i s method of comparison to f u n c t i o n s d e f i n e d on some neighbour-hood of a n o n - i s o l a t e d point of a Hausdorff space and with ranges i n an a r b i t r a r y Banach space. It i s shown, i n Chapter 3» that when a p p l i e d to the elements of a l i n e a r space of f u n c t i o n s , the r e s u l t s of t h i s method can be expressed by a s s i g n i n g to each f u n c t i o n a norm under which the space i s a V-space. L i n e a r spaces s a t i s f y i n g the d e f i n i n g p r o p e r t i e s of a V-space, except f o r the r e t e n t i o n of (O.l) i n place of ( 0 . 3 ) , have been s y s t e m a t i c a l l y i n v e s t i g a t e d by A. F„ Monna [24], [25]. Most of the r e s u l t s of Monna are v a l i d under the a d d i t i o n a l " c b n d i t i o n s that the space be separable or l o c a l l y compact. Except i n t r i v i a l cases, V-spaces are n e i t h e r l o c a l l y compact nor separable. In Chapters 1 and 2 we i n v e s t i g a t e the b a s i c t o p o l o g i c a l and a l g e b r a i c p r o p e r t i e s of V-spaces. A notion of utmost importance i n t h i s work i s that of w d i s t i n g u i s h a b i l i t y " . " D i s t i n g u i s h e d sets 1* and " d i s t i n g u i s h e d bases'* are d e f i n e d i n S e c t i o n 7 , Chapter 1. The concept of d i s t i n g u i s h a b i 1 i t y has been i n t r o d u c e d by Monna [24, V], [25-, I] under a d i f f e r e n t name and through another formal d e f i n i t i o n (see S e c t i o n 2 - 5 ) . Monna has shown that i n non-Archimedean normed l i n e a r spaces over a f i e l d with a n o n - t r i v i a l v a l u a t i o n , d i s t i n g u i s h e d bases e x i s t only under r e s t r i c t i v e con-d i t i o n s . However, by use of a modified form (Theorem 1 -6 . 1 ) of the c l a s s i c a l Riesz's Lemma ( [ 7 ] , [ 3 6 ] ) , i t i s proved i n Theorem 3 2 - 2 . 2 that a V-space admits a d i s t i n g u i s h e d b a s i s . It f o l l o w s (Theorem 1-7=6) that an element belongs to the space i f and only i f i t i s a sum of a formal s e r i e s i n terms of the elements of a d i s t i n g u i s h e d b a s i s . Thus, the r o l e of a d i s t i n g u i s h e d b a s i s in a V-space i s s i m i l a r to the r o l e of a complete orthogonal b a s i s i n a H i l b e r t space. We a l s o c o n s i d e r V-algebras and give theorems on the e x i s -tence of i n v e r s e s and on the s p e c t r a of elements of a V-algebra. Most of these theorems are simple m o d i f i c a t i o n s of the c l a s s i c a l theorems of the theory of normed r i n g s ( [ 7 ] , [ 2 6 ] ) . Examples of V-spaces and V-algebras are d i s p l a y e d i n Chapter 3 . "Asymptotic spaces" are c o n s t r u c t e d by widening the scope of the method of C. A. Swanson and M. S c h u l z e r [ 3 2 ] , [ 3 3 ] , r e f e r r e d to above. We a l s o d e f i n e "moment spaces" i n which, f o r example, one can i n t e r p r e t the methods of Lanczos [21] or Clenshaw [ 2 ] f o r the approximation of the s o l u t i o n s of c e r t a i n d i f f e r e n t i a l equa-t i o n s . Chapter 4 i s devoted to the study of l i n e a r and n o n - l i n e a r operators on V-spaces. By s e t t i n g a proper norm ( D e f i n i t i o n l . l ) on these o p e r a t o r s , the set of bounded operators forms a V-space of which the set of bounded l i n e a r operators i s a subspace (Theorems 2 . 1 , 3 . 1 ) . Elementary theorems (e.g. Theorems 3»3» 3 ° 4 ) of the theory of bounded l i n e a r operators on Banach spaces s t i l l apply in V-spaces. However, important d i f f e r e n c e s are e x e m p l i f i e d : a continuous l i n e a r operator i s not n e c e s s a r i l y bounded (p. 8 6 ) ; , the uniform boundedness theorem does not h o l d (p. 8 7 ) . 4 Theorem 4-1 g i v e s a simple c h a r a c t e r i z a t i o n of bounded l i n e a r o p e r a t o r s . As a p p l i c a t i o n s of t h i s important theorem we d e r i v e a r e s u l t of H. F. Davis [ 4 ] and i n d i c a t e how asymptotic expansions of the Laplace transforms of c e r t a i n f u n c t i o n s of two v a r i a b l e s can be obtained (see V. A. D i t k i n and A. P Prudnikov [ 6 ] ) . Theorem 6.5 allows the comparison of the s p e c t r a of two bounded l i n e a r operators when the norm of t h e i r d i f f e r e n c e i s l e s s than 1. The r e s u l t i s obtained by showing that an i n e q u a l -i t y proved by C. A. Swanson [ 3 4 ] / [ 3 5 ] f o r l i n e a r transforma-t i o n s with eigenvalues on a H i l b e r t space can be m o d i f i e d i n t o an e q u a l i t y i n V-spaces. The problem c o n s i d e r e d by C. A. Swanson and M. S c h u l z e r i n [ 3 2 ] and [ 3 3 ] i s that of the e x i s t e n c e and approximation of "asymptotic s o l u t i o n s " of c e r t a i n equations i n Banach spaces. In- Chapter 5 , we extend the r e s u l t s of Swanson and S c h u l z e r to a r b i t r a r y V-spaces and V-algebras (Theorems 2 . 2 , 3 « 2 , 4 •> 3 ) » Our methods of proof are d i f f e r e n t than those of [ 3 2 ] and [ 3 3 ]• Our hypotheses are weaker and consequently our proofs are more i n -v o l v e d . P o s s i b l e s i m p l i f i c a t i o n s of the hypotheses are mentioned. In Chapter 6 we c o n s i d e r continuous l i n e a r f u n c t i o n a l s . It i s known that continuous l i n e a r f u n c t i o n a l s on a V-space are bounded (Monna [ 2 4 , H I ] ) and that the Hahn-Banach Theorem i s v a l i d (Monna [ 2 4 , I I I ] , Cohen [ 3 ] , Ingleton [ 1 7 ] ; we give a new proof of the l a t t e r u s i n g d i s t i n g u i s h e d bases. The main r e s u l t of t h i s c h a p t e r ' i s a r e p r e s e n t a t i o n theorem (Theorem 3*5) f o r l i n e a r f u n c t i o n a l s on c e r t a i n bounded V-spaces. The r e p r e s e n t a t i o n theorem i s a g e n e r a l i z a t i o n of a theorem of Ho P' o Davis [ 4 ] which a s s e r t s that the space of continuous l i n e a r f u n c t i o n a l 3 on the space of a s y m p t o t i c a l l y convergent power s e r i e s i n a r e a l v a r i a b l e i s isomorphic to the space of p o l y -nomials i n that v a r i a b l e . I t i s shown (Section 6-2) that a new norm, c a l l e d "•norm", ( D e f i n i t i o n 2.2) can be d e f i n e d on the set of f i n i t e l i n e a r combinations of the elements of a d i s t i n g u i s h e d b a s i s of a V-space X, and that,, under t h i s norm, t h i s set i s a V-space isomorphic to a subspace of the dual of X. T h i s isomorphism i s i s o m e t r i c and i s obtained by use of a p a r t i c u l a r type of inner product ( D e f i n i t i o n 3=l)° In Chapters 5 and 6, a p p l i c a t i o n s of the theorems are shown us i n g some of the examples of asymptotic spaces des-c r i b e d i n Chapter 3° 6 CHAPTER 1 "VALUED SPACES 1-1 D e f i n i t i o n s and n o t a t i o n s In t h i s chapter X denotes a l i n e a r space over a f i e l d of s c a l a r s F. F i s a f i e l d with c h a r a c t e r i s t i c 0, i . e . a f i e l d which c o n t a i n s the set of the r a t i o n a l numbers as a s u b f i e l d . The a d d i t i v e i d e n t i t y (zero-element) of X w i l l be denoted by 6 and that of F by 0, D e f i n i t i o n 1 . 1 . X w i l l be c a l l e d a pseudo-valued space i f there e x i s t s a non-negative r e a l valued f u n c t i o n d e f i n e d on a l l of X, whose value at x w i l l be c a l l e d the norm of x and denoted by Jx|„ and which s a t i s f i e s : ( 1 . 1 ) = G, ( 1 . 2 ) iaxj = fx| f o r a l l x ^ X and a l l a £ F, a # 0, ( 1 . 3 ) |x + y | < Max {|x|, l y | } f o r a l l x, y e X. Def i n i t i on 1 . 2 . A pseudo-valued space X w i l l be c a l l e d a s t r o n g l y pseudo-valued space i f f o r a l l x, y £ X, | x | i= | y | i m p l i e s ( 1 . 4 ) |x + y | = Max [|x|, | y j } . D e f i n i t i o n 1 . 3 . A ( s t r o n g l y ) pseudo-valued space X w i l l be c a l l e d a ( s t r o n g l y ) valued space i f ( 1 . 5 ) |x| = 0 i m p l i e s x = 9. C o n d i t i o n ( 1 . 3 ) i s stronger than the u s u a l t r i a n g l e i n e q u a l i t y , s i n c e 7 Max {|x|, |y |} < |x| * |y|. A (pseudo-) valued space i s a (pseu do-) normed l i n e a r space i f F i s understood to have the t r i v i a l v a l u a t i o n 101 = 0, | ct | = 1 f o r a l l a g F, a ^ 0. With t h i j v a l u a t i o n on F, (1.2) can be w r i t t e n as |a x | = | o l | | x | f o r a l l x £ X and a l l a £ F. On a (pseudo-) valued space X a (pseudo-) m e t r i c i s d e f i n e d by d(x, y) = |x - y |. Th i s (pseudo-) m e t r i c i s non-Archimedean, i . e . (1.6) d(x, y) < Max {d(x, z ) , d(z, y)}, x, y, z € X. If X i s s t r o n g l y (pseudo-) valued, the (pseudo-) m e t r i c i s s t r o n g l y non-Archimedean, i . e . f o r a l l x, y, z ^ X, (1.7) d(x, y) = Max {d(x, z ) , d(z, y)} when d(x, z) t d(z, y ) . Every t r i a n g l e i n X i s i s o s c e l e s with the two longest s i d e s being of equal l e n g t h . Indeed, i f d(x, y) > d(y, z) > d(z, x), i t f o l l o w s from (1.6) that d(x, y) = d(y, z ) . The f u n c t i o n d i s t r a n s l a t i o n i n v a r i a n t , i . e . d(x + z, y + z) = d(x, y ) . Examp 1 e. Let A. »c [0, l ] and co n s i d e r the set of r e a l valued f u n c t i o n s ^ = {«pr : 0 < r < »}, cpr(A.) = X,r. 8 Consider the l i n e a r space X c o n s i s t i n g of the zero f u n c t i o n and a l l formal s e r i e s of the form x = a cp + a.» + a o Cp + a, r e a l , a f 0, oTr l T r , 2 T r _ ' 1 ' o ' ' o 1 2 where r ^ _ ^ < r^ f o r i = 1, 2 , and where the set [ r Q , r ^ , •••} i s e i t h e r f i n i t e or i s i n f i n i t e and unbounded. Let o be a f i x e d r e a l number, 1 < p < and d e f i n e on X the f u n c t i o n - r 101 = 0, |x| = p ° i f x = a otp r + °°° . o One v e r i f i e s e a s i l y that t h i s f u n c t i o n s a t i s f i e s ( l . l ) , ( 1 . 2 ) , ( 1 . $ ) . Thus, X i s a st rongly valued space. The subset Y of X c o n s i s t i n g of 0 and of a l l those p o i n t s x f o r which {*"_,, ri» •••} i s a set of r a t i o n a l numbers i s a l i n e a r subspace of X. S i m i l a r l y the subset of X c o n s i s t i n g of 0 and of a l l those p o i n t s x f o r which { r Q , r ^ , •••} i s a set of i n t e g e r s i s a sub-space of X. Other examples are c o n s t r u c t e d in Chapter 3, Remark on the terminology. The word " v a l u e d " was i n t r o d u c e d to avoid the heavy l o c u t i o n s which would have r e s u l t e d from the use of the g e n e r a l l y accepted terminology. A " s t r o n g l y pseudo-v a l u e d " space i s a " s t r o n g l y non-Archimedean pseudo-normed l i n e a r space over a f i e l d with a t r i v i a l v a l u a t i o n " (!). The word " v a l u e d " i s meant to r e c a l l the p a r t i c u l a r v a l u a t i o n which i s imposed upon the f i e l d of s c a l a r s as w e l l as the s i m i l a r i t i e s of 9 the d e f i n i n g p r o p e r t i e s ( l . l ) , ( l » 3 ) with those of a v a l u a t i o n i n g e n e r a l ( [ 3 l ] f [AO]). 1-2 Some t o p o l o g i c a l p r o p e r t i e s In t h i s s e c t i o n we s h a l l l i s t some of the t o p o l o g i c a l p r o p e r t i e s of a (pseudo-) valued space X. For the c l a s s i c a l terminology we r e f e r to textbooks on topology or a n a l y s i s (e.g. [ 7 ] ; [ 1 8 ] ; [19], V o l . I; [ 3 6 ] ) . The topology c o n s i d e r e d i s the topology induced on X by the (pseudo-) m e t r i c d of 1-1; we r e c a l l that t h i s topology i s the smallest topology which c o n t a i n s a l l the b a l l s S(x, r ) , x € X, r > 0. The open b a l l S(x, r ) , the c l o s e d b a l l S'(x, r) and the sphere B(x, r ) , with center x and rad i u s r, are d e f i n e d by (1.8a) S(x, r) = {y e X : d(x, y) < r}, r > 0 , (1.8b) S»(x, r) = (y * X : d(x, y) < r } , r > 0 , (1.8c) B(x, r) = {y g X : d(x, y) = r}, r > 0 . ( i ) S ince the (pseudo-) m e t r i c d i s t r a n s l a t i o n i n v a r i a n t ( l - l ) the neighbourhood system of a point x i s the x - t r a n s l a t e of the neighbourhood system of 0, i . e . i f V i s a neighbourhood of x and W i s a neighbourhood of 9, then V - x i s a neighbourhood of 9 and W + x i s a neighbourhood of x. ( i i ) For any x, y € X and r, S(x, r) • S(y, r) or S(x, r) fl S(y, r) = 0 S»(x, r) =S»(y, r) or S»(x, r) n S'(y, r) = 0 10 where 0 denotes the empty s e t . To prove the f i r s t statement i t i s s u f f i c i e n t to see that i f z € S(x, r) n S(y, r) and u € S (x, r) then u € S(y, r ) . Indeed, d(y, u) < Max fd(y, z ) , d(z, u)} < Max {d(y„ z ) , d(z, x ) , d(x, u)} < r . The second statement i s proved i n a s i m i l a r way. ( i i i ) If X i s a s t r o n g l y (pseudo-) valued space, then f o r any x € X and r > 0, S(x, r ) , S f ( x , r ) , B(x, r) are a l l c l o s e d and open. The proofs.for S(x, r) and S'(x, r) are s i m i l a r . For B(x, r) i t f o l l o w s from the e q u a l i t y B(x, r) = S* (x, r ) \ s ( x , r ) . That S(x, r) i s open i s guaranteed by the d e f i n i t i o n of the topology induced on X by d. To show that i t i s c l o s e d , l e t y £ X \ s ( x , r ) . For every z £ S (y, - | r ) , d(z, y) < d(x, y ) , and, by (1.7), d(z, x) > r . Thus X^s vS(x, r) c o n t a i n s a neighbour-hood of each of i t s p o i n t s ; i t i s open, and S(x, r) i s c l o s e d . ( i v ) A component of a t o p o l o g i c a l space i s a maximal connected subset of the space ([18], p. 54)« It i s a consequence of ( i i i ) that i f X i s s t r o n g l y valued, then each component of X c o n s i s t s of a s i n g l e p o i n t ; i f X i s 11 s t r o n g l y pseudo-valued each component i s a t r a n s l a t e of [ 0 ] , where [9] = S» (9, 0) = U £ X : | x | = O l . Spaces whose components c o n s i s t of s i n g l e p o i n t s are c a l l e d t o t a l l y d i s c o n n e c t e d ([28], p. 76). (v) A space X i s s a i d to be 0-dimensional i f f o r any x ^ X every neighbourhood of x c o n t a i n s a neighbourhood of x whose boundary i s empty ([16], pp. 10, 1 5 ) . It f o l l o w s from ( i i i ) that a s t r o n g l y (pseudo-) valued space i s 0-dimensional, s i n c e the b a l l s S(x, r) have empty boundaries. It has been proved by J . de Groot ( [ 5 ] , Th. II) that a m e t r i z a b l e space admits a non-Archimedean metric i f and only i f i t i s s t r o n g l y 0-dimensional. In de G r o o t T s terminology, a space i s s t r o n g l y 0-dimensional i f and only i f i t i s a Hausdorff space and admits a < - - l o c a l l y f i n i t e b a s i s f o r i t s topology c o n s i s t i n g of subsets which are bot,h c l o s e d and open.* In the case of a s t r o n g l y valued space X, the f a m i l y A = U{__ : r > 0, r r a t i o n a l j where _ r = l s ( x , r) : x € X, r > 0} forms a (--locally f i n i t e b a s i s . Indeed $ i s a b a s i s , i t i s the * The terminology i s that of K e l l e y ([18], pp. 126, 1 2 7 ) . A f a m i l y of sets i s c a l l e d c - l o c a l l y f i n i t e i f i t i s the union of a countable number of l o c a l l y f i n i t e s u b f a m i l i e s . A f a m i l y of sets i s c a l l e d l o c a l l y f i n i t e i f every point has a neighbourhood which i n t e r s e c t s at most a f i n i t e number of sets of the f a m i l y . 12 countable union of the &r' s and, f o r a f i x e d r, the f a m i l y i s l o c a l l y f i n i t e , by ( i i ) above. (For a given x £ X, S(x, r) i n t e r -s e c t s only one set i n % r'- i t s e l f . ) Other e q u i v a l e n t d e f i n i t i o n s of O-dimensionality and i t s consequences are s t u d i e d i n [16], Ch. I I . ( v i ) A f i e l d F forms a s t r o n g l y valued space over i t s e l f i f the norm f u n c t i o n i s i d e n t i c a l to the t r i v i a l v a l u a t i o n : (1.9) | 0 | = 0 and | a | = 1 f o r a l l a € F, a + 0. The topology induced on F by t h i s norm i s the d i s c r e t e topology on F ([18], p. 37).' In the sequel, whenever the f i e l d F i s the f i e l d R of the r e a l numbers or the f i e l d C of the complex numbers, the symbols (1.9) w i l l be r e t a i n e d to denote the t r i v i a l value of the numbers. The symbol |a| w i l l denote the usual absolute value of a, i . e . |a| =J~G? if a. g R and |a| = | a + b i | =J a 2 + b 2 if a C C. The t o p o l o g i e s induced on R and C by t h e i r u s ual v a l u a t i o n s w i l l be c a l l e d the u s u a l t o p o l o g i e s on R and C. We conclude t h i s s e c t i o n by the f o l l o w i n g Theorem 2.1. If X i s a s t r o n g l y (pseudo-) valued space and r > 0, then ( i ) S(0, r) and S'(9, r) are subspaces of X; ( i i ) The q u o t i e n t t o p o l o g i e s on the q u o t i e n t spaces X //S(9, r) and X / S » ( 9 , r) are both d i s c r e t e . 13 Proof: ( i ) i s e a s i l y v e r i f i e d . ( i i ) For the terminology, we r e f e r to [28], pp.- 59, 60. The n a t u r a l mapping from a t o p o l o g i c a l group to one of i t s q u o t i e n t groups i s a continuous open mapping. The p o i n t s i n the quotient groups X / S ( 9 , r) and X / S ' ( 9 , r) are t r a n s l a t e s of the b a l l s S(0, r) and S'(9, r) r e s p e c t i v e l y . The b a l l s S(9, r) and S»(0, were shown to be both c l o s e d and open (see ( i i i ) above), thus th p o i n t s i n the q u o t i e n t groups are both c l o s e d and open ([28], p. over the f i e l d of s c a l a r s F. Theorem 3.1. Let f be a continuous f u n c t i o n from D to X, where D i s an a r b i t r a r y t o p o l o g i c a l space. Let g be the f u n c t i o n with domain F x D d e f i n e d by: g(a, u) | a f ( u ) | , a e F, u e D. If v € D i s such that | f ( v ) | r" 0, th en there e x i s t s a neighbourhood W(v) of v such that f o r a l l u € W(v) g(P, u) = g ( l , v) = | f (v) |, 0 « F, p r o v i d e d |3 r" 0. Proof: If M 0 then g(3, u) «= 0 f | f ( v ) | . If j3 f 0, then by the c o n t i n u i t y of f, there e x i s t s a neighbourhood W(v) of v such that f o r a l l u £ W(v) 5 9 ) . 1-3 Some p r o p e r t i e s of the norm f u n c t i o n In t h i s s e c t i o n , X i s a s t r o n g l y (pseudo-) valued space ] f (u) - f ( v ) | < [ f ( v ) | 0 . By ( l . A ) , t h i s i m p l i e s that | f ( _ ) | = | f ( v ) | f o r a l l u C W(v). By ( 1 . 2 ) , s i n c e 0 * 0 , g(3, u) = | {3f (u) | = | f ( u ) | = l f ( v ) | f o r a l l u £ W(v). If we l e t a = 1 i n the theorem, we o b t a i n : C o r o l l a r y 3.2. Let f be a continuous f u n c t i o n from D to X, where D i s an a r b i t r a r y t o p o l o g i c a l space. If v £ D i s such that | f ( v ) | ir"' 0 , then | f ( u ) | = | f ( v ) | f o r a l l u i n some neighbourhood of v. In Theorem 3 - 1 , l e t D ='. X and f be the i d e n t i t y f u n c t i o n . We obt ain : C o r o l l a r y 3.3. Let F = R or F = C. Let F x X have the c r o s s -product topology induced by the usual topology on F and the topology on X ( [ 18 ] , p. 9 0 ) . ( i ) If (a, x) £ F x X, a £ 0 and |x| * 0 , then there e x i s t s a neighbourhood V of (a, x) i n F x X, such that f 0y 1 = | x | f o r a l l (P, y) cc V. ( i i ) The f u n c t i o n g(d, x) = |ax|, d e f i n e d on F X X, i s d i s c o n t i n u -ous at (a, x) i f a = 0 and | x | r" 0 ; i t i s continuous at a l l other p o i n t s . Remark: The c o n c l u s i o n of C o r o l l a r y 3«3 remains true f o r an a r b i t r a r y f i e l d F with c h a r a c t e r i s t i c 0 and with the topology induced by some v a l u a t i o n ( [ 4 0 ] , Ch. X; [ 3 1 ] , Ch. 2), i f the 15 r e s t r i c t i o n of t h i s v a l u a t i o n to the s u b f i e l d of the r a t i o n a l numbers i s i d e n t i c a l to the usual v a l u a t i o n of the r a t i o n a l s . This remark a l s o a p p l i e s to the statements ( i i i ) and ( i v ) below. The f o l l o w i n g p r o p e r t i e s of a s t r o n g l y (pseudo-) valued space X can be v e r i f i e d d i r e c t l y or by use of the l a s t c o r o l l a r y ; ( i ) If x € X and |x| f 0, th en the subspace generated by x, with i t s r e l a t i v e topology, i s a d i s c r e t e t o p o l o g i c a l space. The d i s t a n c e between any p a i r of d i s t i n c t p o i n t s , ax and j3x, i s con-stant and equal to |x|. ( i i ) If F i s given the d i s c r e t e topology, then the f u n c t i o n f : f ( a , x) = ax, d e f i n e d on F x X, i s continuous i n a f o r a f i x e d x and i s continuous i n x f o r a f i x e d a. ( i i i ) Let F = R or F = C and l e t F have i t s usual topology. The f u n c t i o n f : f ( a , x) = ax, d e f i n e d on F X X, i s continuous i n a, f o r a f i x e d x, i f and only i f | x | = 0; i t i s continuous i n x f o r a f i x e d a. ( i v ) Let F = R or F = C and l e t i ^ } be a sequence of d i s t i n c t s c a l a r s convergent to a i n the usual topology of F. The sequence l a n x l converges to ax i n the topology of X, i f and only i f 1*1 = o-(v) If l x n ) i s a sequence i n X, convergent to a l i m i t x such that | x | i= 0, then every x^ £ S(x, |x|) i s 'such that f x n f = |x|. Fur-ther, there are at most a f i n i t e number of i n d i c e s n such that x = a x, a e F and a £ 1. n n n v n 16 1-4 Convergence of sequences and s e r i e s In t h i s s e c t i o n two important theorems concerning the con-vergence of sequences and s e r i e s i n (pseudo-) valued spaces w i l l be s t a t e d . For the c l a s s i c a l terminology, we r e f e r to [ 7 ] , P« 19; [ 1 9 ] , V o l . I, p. 3 6 ; or [ 3 6 ] , p. 74-Theorem 4 ° 1 . If X i s a (pseudo-) valued space: ( i ) A sequence I x n } i n X i s a Cauchy sequence i f and only i f l i m d(x , x .) = l i m |x - x .1 = 0. n n+l • n n+l" n - t m n«*ao ( i i ) A s e r i e s ^ n x n i Q ^  i s a Cauchy s e r i e s i f and only i f l i m d(x , 9) = lim 1x1=0. n' . . • n * n -»oo n mna> The proof of t h i s theorem i s omitted. It i s a mere modifica-t i o n of the proof of a s i m i l a r theorem f o r f i e l d s with a non-Archimedean v a l u a t i o n . See [ 3 1 ] , P- 28 or [ 4 0 ] , P- 2 4 0 . The proof of the f o l l o w i n g theorem i s al s o omitted ( c f . Lemma 7 .5 below). Part ( i ) i s quoted, without proof, i n [ 2 9 ] , p. 139- Part ( i i ) f o l l o w s from Theorem 1.5 ( i i ) and i n e q u a l i t y ( 1 . 3 ) . Theorem 4 . 2 . If X i s a (pseudo-) valued space: ( i ) A convergent s e r i e s i s u n c o n d i t i o n a l l y convergent, i . e . any r e o r d e r i n g of i t s terms converges to the same sum(s). ( i i ) If n x n i s convergent and has sum x, then | x | < sup |x n|. 17 1-5 Compactness In t h i s s e c t i o n we give a c h a r a c t e r i z a t i o n of the compact subsets of a s t r o n g l y (pseudo-) valued space X. For d e f i n i t i o n s and p r o p e r t i e s r e l a t e d to compactness we r e f e r t o textbooks (e.g. [ 7 ] ; [18]; [ 1 9 ] , V o l . I; [ 3 6 ] ) . If the topology of X i s d i s c r e t e , a subset of X i s compact i f and only i f i t i s f i n i t e . Thus X i t s e l f i s not compact. Since each p o i n t forms a neighbourhood of i t s e l f , X i s l o c a l l y compact. If the topology of X i s not d i s c r e t e , then X i s n e i t h e r com-pact nor l o c a l l y compact. Indeed every neighbourhood V of 0 con-t a i n s a b a l l S(9, r) f o r some r . This b a l l c o ntains a point x such that | x | £ 0 . Thus V contains the d i s c r e t e subspace generated by x ( ( i ) , page 15.). We s h a l l use the f o l l o w i n g d e f i n i t i o n : D e f i n i t i o n 5 . 1 . ' Let A c X. The s e t Q ( A ) d e f i n e d by H ( A ) = {r : | x f = r f o r some x e At w i l l be c a l l e d the norm range of A. Theorem 5 . 2 . Let X be a s t r o n g l y (pseudo-) valued space, and A be a subset of X. ( i ) A i s compact i f and only i f f o r each r > 0 i t i s a f i n i t e union of d i s j o i n t compact subsets K^, K.^, K n ( r ) , such that x g and y £ i m p l i e s |x - y | < r f o r i = j , ( 1 . 1 0 ) |x — y | > r f o r i f j . 18 ( i i ) If A i s compact and does not c o n t a i n 9, except p o s s i b l y as an i s o l a t e d p o i n t , then i t s norm range Q ( A ) i s f i n i t e . Proof; ( i ) A set i s c e r t a i n l y compact i f i t i s a f i n i t e union of compact s e t s . For the converse, l e t A be compact and r > 0 be a r b i t r a r y . The f a m i l y A = {S(x, r) : x £ AJ i s an open cover of A. We can e x t r a c t from & a f i n i t e subcover {S(x 1, r ) , S ( x 2 , r ) , S ( x n ^ y r)} such that the S(x., r) are d i s j o i n t . (See ( i i ) , page 9)• Then (1.10) i s s a t i s f i e d with r e p l a c e d by S(x^, r ) . Take = A n S(x^, r ) . i s compact since i t i s the i n t e r s e c t i o n of a compact set and a c l o s e d set ( ( i i i ) , page 10). Then (1.10) h o l d s . ( i i ) S i n c e 9 i s at most an i s o l a t e d p o i n t of A, there e x i s t s r > 0 such that |x| > r f o r a l l x £ A, x f 9. Consider, f o r t h i s p a r t i c u l a r value of r, the sets , i = 1, 2, n ( r ) of ( i ) . Then, 9 £ f ^ , x £ and y £ K., imply I x I > r > | y | > r and | x - y | < r . Since X i s s t r o n g l y (pseudo-) valued, by (1.4), |x| = | y | and the c o n c l u s i o n f o l l o w s . Remarks : ( i ) The f a c t that the v a l u a t i o n on the f i e l d F 19 i s the t r i v i a l v a l u a t i o n i s r e s p o n s i b l e f o r a high d i s c r e t i z a t i i n a (pseudo-) valued space. As a r e s u l t , we may say, l o o s e l y speaking, that compactness i s a very r e s t r i c t i v e property and that i t i s r a t h e r d i f f i c u l t f o r a subset of a (pseudo-) valued space to be compact. No convex set i s compact unless i t i s reduced to a s i n g l e point (or to a subset of S(9, 0 ) ) . No set with a non-empty i n t e r i o r i s compact unless the space i s d i s c r e t e and the set i s f i n i t e . One can expect that compactness w i l l not play an important r o l e i n t h i s t h e o r y . ( i i ) The r e s u l t s of Theorem 5 . 2 may be compared with Property 4 , i n Theorem 2 of Monna, [ 2 4 ] , Part I, page I O 4 8 . Monna has shown that i f a non-Archimedean normed l i n e a r space over a f i e l d of s c a l a r s with the t r i v i a l v a l u a t i o n i s l o c a l l y compact, then the f i e l d of s c a l a r s i s f i n i t e . ( [ 2 4 ] , Part I I , p. 1061.) 1-6 M o d i f i c a t i o n of Reisz's Lemma One can expect that many theorems i n the c l a s s i c a l theory of normed l i n e a r spaces w i l l have analogues i n the theory of valued spaces. The theorem of t h i s s e c t i o n i s given as an example of a m o d i f i e d statement and i t s p r o o f . In the case of a normed l i n e a r space X over the r e a l or complex f i e l d with the us u a l v a l u a t i o n s , R i e s z ' s Lemma can be .20 s t a t e d as f o l l o w s ( [ 3 6 ] , p. 9 6 ; a l s o [ 7 ] , P. 5 7 8 ) : "Let Y be a c l o s e d , proper subspace of X. Then f o r each a such that 0 < a < 1, there e x i s t s a point x a c X such that f | x j = 1 and ||y - xj > a f o r a l l y € Y." If X i s a s t r o n g l y (pseudo-) valued space, the above s t a t e -ment must be m o d i f i e d . The reason f o r the a l t e r a t i o n i s the i m p o s s i b i l i t y of n o r m a l i z i n g an element i n X, i . e . the impossi-b i l i t y of f i n d i n g , f o r each x such that |x| * 0, a s c a l a r <x such that |ax| = 1 (unless, of course, |x| = 1 f o r a l l x £ X such that Theorem 6 . 1 . (Modified R i e s z ' s Lemma) Let Y be a c l o s e d , proper subspace of a s t r o n g l y (pseudo-) valued space X. For each a such that 0 < a < 1, there e x i s t s a point x £ X such that |x| * 0 ) . |y - x a | > a | x a | f o r a l l y € Y. Proof: ( i ) If there e x i s t s z £ X, | z | £ 0, such that |y - 21 > I z I f o r a 1 1 y € Y , t ake x a = z f o r a l l a, 0 < a < 1. Define 6(y) - y € Y, | y | r- 0. 6 = i n f 6(y) y€Y 21 Then, 6 < 6 ( y Q ) < 1 . Moreover 6 (y) < 1 i m p l i e s |y - X q | < | y | and hence | y | = | X q | ; t h e r e f o r e , s i n c e Y i s c l o s e d , 6 (y) = — j — ^ — | — i s bounded away from zero f o r y € Y ° Thus 0 < 6 < 1 . Let a be given, 0 < a < 1, and l e t 6 ' = min { a - 1 6 , -| (1 + 6 ) } . 6 < 6 ' < 1 . There e x i s t s y^ € Y such that | y, - x \ 6 ( y i } = f x : | ° < • o 8 Let x,v = x - y •, . Then cx o l ( 1 . 1 1 ) | x a | < 6 > | x J = 6 ' l y J . Now l e t y £ Y. If (y - x a | > | x a | the proof i s f i n i s h e d , so we may assume h ~ x a l < l * a l ' Then | y | = | x a | < |y x | (by ( l . l l ) ) so that |y + y j = l y j . Hence s i n c e y •+ y^ £ Y, we have by the d e f i n i t i o n of 6 : |y - x a | = \y * y x - x Q J > 6|y + y j = 6%y J , and by ( l . l l ) l y " x a l > a I * a l > a K « -T h i s completes the p r o o f . Example. Consider the s t r o n g l y valued space X of the Example of page 7 and i t s subspace Y (page 8 ) . 22 To v e r i f y that Y i s c l o s e d , l e t x £ X\Y; then x • f 3 o ? s o + * r ? 8 l + • P o * °< where f o r some i n t e g e r p, 3 * 0 , s i s i r r a t i o n a l and s. i s P P 1 ^ -s r a t i o n a l f o r a l l i , 0 < i < p. C l e a r l y | x | > p p. If y £ Y, y f 0 , we have y = a m + a m + •»•, a * 0 , r, r a t i o n a l f o r a l l i . VP'o T l ^ i T h e r e f o r e , j x - y | > -s P s i n c e X _ y = • • « + BfQ + » • • pTs , P Thus, no sequence i n Y can converge to a poin t of X \ Y ; Y i s a c l o s e d proper subspace of X. Given a, 0 < a < 1, l e t s be i r r a t i o n a l and a < s < 1. The same argument as above shows that I T - «p 8| > l<p sl > a|<ps.| f o r a l l y 6 Y . 1-7 D i s t i n g u i s h e d bases In the p r e v i o u s s e c t i o n s we were concerned with p r o p e r t i e s of valued spaces which were mostly of t o p o l o g i c a l nature. In t h i s s e c t i o n we i n t r o d u c e a l g e b r a i c concepts which depend on l i n e a r i t y . We r e c a l l a few c l a s s i c a l d e f i n i t i o n s ( [ 7 ] , pp. 3 6 , 4 6 , 5 0 ; [ 3 6 ] , pp. 4 4 , 4 5 ) . Let A be a subset of a t o p o l o g i c a l l i n e a r space X, over a f i e l d F. The subspace (A) generated by A i s the set of a l l the f i n i t e l i n e a r combinations of elements of A. The t o p o l o g i c a l c l o s u r e of (A) w i l l be denoted by [A] and be c a l l e d 23 t h e c l o s e d s u b s p a c e g e n e r a t e d by A. The s e t A i s s a i d t o be l i n e a r l y i n d e p e n d e n t i f f o r any f i n i t e s u b s e t {x^ # x^, °*°, x n} of A, 1 1 2 2 n n ' i ^ ' i m p l i e s = 0 f o r a l l i . A i s c a l l e d a Hamel b a s i s of X i f A i s a l i n e a r l y i n d e p e n d e n t s e t and (A) = X. I t i s known ( [ 7 ] , [ 3 ° ] ) t h a t a l i n e a r s p a c e has a Hamel b a s i s ; t h a t a s u b s e t i s a Hamel b a s i s i f and o n l y i f i t i s a maximal l i n e a r l y i n d e p e n d e n t s e t ; t h a t a l l t h e Hamel b a s e s of a space X have t h e same c a r d i n a l i t y . We add t h e f o l l o w i n g d e f i n i t i o n : D e f i n i t i o n 7 . 1 . L e t A be a s u b s e t of a t o p o l o g i c a l l i n e a r s p a c e X. ( i ) A i s s a i d t o be a c o m p l e t e l y i n d e p e n d e n t s e t i f x £ L A \ i x l ] f o r each x £ A. ( i i ) A i s c a l l e d a c o m p l e t e b a s i s i f i t i s a c o m p l e t e l y i n d e p e n -dent s e t and [A] = X. C l e a r l y a c o m p l e t e l y i n d e p e n d e n t s e t i s a l s o l i n e a r l y i n d e p e n -d e n t . The c o n v e r s e i s not t r u e as i s shown by t h e f o l l o w i n g e xample. L e t C [ 0 , l ] be t h e space of a l l t h e r e a l v a l u e d c o n t i n u -ous f u n c t i o n s f on [ 0 , l ] , w i t h t h e u n i f o r m norm: fjfjj = sup | f (A.) | ; l e t X be t h e s u b s p a c e of C [ 0 , l ] g e n e r a t e d by 0<A.<1 0 U E 0 , where 24 E o = ( e _ r : r > 0}, e _ r ( \ ) = e " r \ It i s known (Wei e r s t r a s s ' Theorem, [ 3 6 ] , [ 3 7 ] ) that the set $ o i s a completely independent set which i s a complete b a s i s but not a Hamel b a s i s of X. Thus $ q (J E e i s a l i n e a r l y independent but not completely independent s e t . Theorem 7.2. A complete b a s i s A of a t o p o l o g i c a l l i n e a r space X i s a l s o a Hamel b a s i s of X i f and only i f (A) co n t a i n s an open set Proof : If A i s a Hamel b a s i s (A) 3 X. Conversely, suppose that (A) c o n t a i n s an open s e t . Then (A) contains an i n t e r i o r point and (A) i s a subgroup of X. It i s known ([18], p. 106) that any sub-group of a t o p o l o g i c a l group which contains an i n t e r i o r point i s c l o s e d (and open). Thus, A i s a l i n e a r l y independent set and (A) = [A] = X. Returning to the theory of valued spaces, we introduce the no t i o n of d i s t i n g u i s h a b i 1 i t y i n the f o l l o w i n g way. D e f i n i t i o n 7.3. Let A be a non-empty subset of a (pseudo-) valued space X. ( i ) A i s s a i d t o be a d i s t i n g u i s h e d set i f no element of A has norm equal t o 0, and, i f f o r any f i n i t e subset of d i s t i n c t p o i n t s x, , x O J • • •, x of A, 1 2 n ' I a i X l + a 2 X 2 + + ttnXnB = M a X 1 X i I' a i £ F, whenever j= 0 f o r i = 1, 2, n. 25 ( i i ) A i s c a l l e d a d i s t i n g u i s h e d b a s i s of X i f A i s a d i s t i n -guished set and a complete b a s i s of X. In s e c t i o n 4 of Chapter 3 , we show that there e x i s t s a norm on the space X of the example of page 2 3 , under which X i s a s t r o n g l y pseudo-valued space. The Hamel b a s i s $ Q \J E q w i l l be shown to be n e i t h e r a complete nor a d i s t i n g u i s h e d b a s i s . The e s s e n t i a l f e a t u r e of a d i s t i n g u i s h e d set A i n a s t r o n g l y (pseudo-) valued space i s the f o l l o w i n g : i f x, y £ A, x * y and l x l = 1 y I = r, then | ax + 3y | = r, except when a = .6 = 0. The author has not been able to show the e x i s t e n c e of d i s -t i n g u i s h e d bases i n a r b i t r a r y (pseudo-) valued spaces. Never-t h e l e s s , under an important a d d i t i o n a l assumption on the norm range of the space., we s h a l l prove, i n Chapter 2, that a s t r o n g l y (pseudo-) valued space has a d i s t i n g u i s h e d b a s i s . T h i s assumption w i l l be s a t i s f i e d i n a l l the examples i n Chapter 3 a n d a p p l i c a -t i o n s i n Chapters 4 , 5 and 6 . In the case of an a r b i t r a r y (pseudo-) valued space, we have: Theorem 7 . 4 . A (pseudo-) valued space admits a Hamel b a s i s which i s a d i s t i n g u i s h e d s e t . The proof i s i d e n t i c a l to the proof- of the e x i s t e n c e of a Hamel b a s i s i n a l i n e a r space ( [ 3 6 ] , p. 4 5 ) . A d i s t i n g u i s h e d Hamel b a s i s i s a maximal d i s t i n g u i s h e d s e t . In the remainder of t h i s s e c t i o n we r e s t r i c t our a t t e n t i o n to s t r o n g l y (pseudo-) valued spaces. In s t r o n g l y (pseudo-) 26 valued spaces, D e f i n i t i o n 7 • 3 ( i i ) i s s l i g h t l y redundant. Indeed, we s h a l l prove i n Theorem 7 « 6 ( i ) below that i n those spaces a d i s t i n g u i s h e d set i s completely independent. Thus a d i s t i n g u i s h e d b a s i s A i n a s t r o n g l y (pseudo-) valued space X i s a d i s t i n g u i s h e d subset such that [A] = X. To prove Theorem 7 » 6 we s h a l l need the f o l l o w i n g lemma, which i s an improvement over Theorem 4 ° 2 ( i i ) . Lemma 7 . 5 ° Let A be a d i s t i n g u i s h e d subset of a s t r o n g l y (pseudo-) v a l u e d space X. Let I x n 1 be an at most countable subset of A. If a 6 F, a * 0 f o r each n, and x = Y a x , then n * ' n ' (j\ n n' |x| = sup | x n | . Proof ; By Theorem 4 . 1 ( 1 1 ) , given r < |x 11, there e x i s t s N such that f o r a l l n > N, |x^ | < r . By Theorem 4 . 2 ( i i ) 1 _ a x I < r, n>N + l and, s i n c e A i s d i s t i n g u i s h e d . I I a n X n l = m a X l X n l ^  I xl I >T° Kn<N n = l — — Thus, N |x| = I Y a x + V a x | = max f x I 1 1 1 L n n - A n n^ • n* . , M Kn<N n=l n>N sup | x n | . n Note t h a t t h i s supremum i s a t t a i n e d . Theorem 7 . 6 . Let X be a s t r o n g l y (p-seudo-) valued space. ( i ) A d i s t i n g u i s h e d subset A of X i s completely independent. 2 7 ( i i ) I f A i s a d i s t i n g u i s h e d b a s i s of X, t h e n e v e r y x £ X can be _ » r e p r e s e n t e d u n i q u e l y ( e x c e p t f o r o r d e r ) by a s e r i e s V a x , n = l w i t h x n g A, a n £ F, n = 1, 2 , • • • . P r o o f ; ( i ) Let x q be an a r b i t r a r y p o i n t of A. Suppose t h a t XQ € L B ] » where B = A s s { x o } . Then t h e r e e x i s t s a s e quence {y } i n (B) w hich c o n v e r g e s t o x . I t f o l l o w s t h a t t h e r e e x i s t s an o e l e m e n t y of (B) y = a.x, + a 0 x 0 + ••• + a x., a, £ 0 , x. e B, 1 ± 2 2 m m ' i . ' i * ' i = 1 , 2, ' • •, m, s uch t h a t |y - x q | < | x q |, i . e . m | Y a x | < fx | < Max f x |„ a = - 1 . , L n " 1 ° 0<i<m 1 ° 1=0 — — T h i s c o n t r a d i c t s t h e d i s t i n g u i s h a b i 1 i t y of A. ( i i ) S i nee x £ [ A ] , t h e r e e x i s t s a s e quence (y : n = 1 , 2 , " ' - } i n (A) w hich c o n v e r g e s t o x. Let z^ = y ^ and z^ = y^ - Yn_^ € (A) f o r n = 2 , 3 , Then t h e s e r i e s ^ z n c o n v e r a / e s t o x. Let z = a 1 x . , + a _ x 0 + »- » + a / \ x , x , a , £ 0 , x . c A , n n l n l N 2 N 2 n , p ( n j n , p ( n j n j ' n j 5 m m. The s e t ixn^ : n = 1, 2 , J = 1, 2, •••, p ( n ) } i s a c o u n t a b l e s e t . Let i t s e l e m e n t s be o r d e r e d i n t o a sequence {x m : m = 1 , 2 , • • • } such t h a t j x m | > | * m + 1 l f o r a l l F o r e a c h i n t e g e r m > 1, t h e r e e x i s t s an i n t e g e r N(m) such t h a t zn l < i x m i f o r a 1 1 n > N ( m ) « 28 T h e r e f o r e , f o r each m, the number of i n t e g e r s n such that x = x , f o r some j , 1 < j < p ( n ) , i s f i n i t e . The s e r i e s m nj — J — *-V 0 0 . z can thus be reordered by grouping the terms i n x , f o r £ n = l n m> each i n t e g e r m. The uniqueness (except f o r order) f o l l o w s from Lemma 7.5» Consequences of the above theorem are: ( i ) If A i s a d i s t i n g u i s h e d subset of X, then Q ( A ) = Q ( ( A ) ) \ { 0 } = Q ( [ A ] ) \ { 0 J . ( i i ) If A i s a d i s t i n g u i s h e d b a s i s of X, then i n f Q ( A ) = 0 when X i s not d i s c r e t e , and i n any case 0 ( A ) = 0 ( X ) \ f o ] , i . e . f o r every r € F _ ( X ) , r * 0 , there e x i s t s x £ A such that | x _ = r . If A i s a d i s t i n g u i s h e d b a s i s of a s t r o n g l y (pseudo-) valued space X, the unique s e r i e s (1.12) Y a n x n , x n € A, a n € F, a n t 0 , n = l which converges to a given p o i n t x £ X, w i l l be c a l l e d the expansion of x i n terms of A. A c c o r d i n g to Theorem 4°2(i), the terms of such an expansion can be reordered to give a non-i n c r e a s i n g s e r i e s , i . e . a s e r i e s such that § x ^ | > | x n + ^ | f o r a l l n > 1. N ot at i on ; If _ ( I and y £ A, we s h a l l denote by (x, y ) ^ the c o e f f i c i e n t of y i n the expansion of x i n terms of A. With t h i s n o t a t i o n (1.12) becomes 29 GO ( 1 . 1 3 ) x = £ (x, x n ) A x n . n = l Assuming that a s t r o n g l y (pseudo-) valued space X admits a d i s t i n g u i s h e d basis,, we can s t a t e Theorem 7 . 7 . A l l d i s t i n g u i s h e d bases of X have the same c a r d i -n a l i t y o This theorem j u s t i f i e s : D e f i n i t i o n 7«8° If a s t r o n g l y (pseudo-) valued space admits a d i s t i n g u i s h e d b a s i s , the c a r d i n a l i t y of t h i s b a s i s w i l l be c a l l e d the ( a l g e b r a i c ) dimens.i on of the space. The proof of Theorem 7 . 7 i s omitted; i t i s s i m i l a r to the proof given by Dunford and^ Schwartz ' (f_7] # P- 253) f o r the i n v a r i -ance of the c a r d i n a l i t y of complete orthonormal bases of a H i l b e r t space.* Theorems 7 . 6 and 7 .7 i n d i c a t e t h a t , to some extent, the r o l e s of d i s t i n g u i s h e d sets and d i s t i n g u i s h e d bases i n a s t r o n g l y (pseudo-) valued space are s i m i l a r to the r o l e s of orthogonal sets and orthonormal bases i n a H i l b e r t space ( [ 7 ] , pp. 2 5 2 - 2 5 3 ) ° * If A and B are two d i s t i n g u i s h e d bases of X, the only m o d i f i c a -t i o n to [ 7 ] , P« 2 5 3 , i s the replacement of the words "... the inner product of a and b i s non-zero...'*, or of the symbol "... (a, b) + 0 ...» by (a, b)Q t 0 and (b, a ) A £ G ..." . 3 0 CHAPTER 2 V-SPACES 2 - 1 D e f i n i t i ons A s y s t e m a t i c s t u d y of n o n - A r c h i m e d e a n normed l i n e a r s p a c e s has been made by A. F. Monna ( [ 2 4 ] , [ 2 5 ] ) ° O t h e r r e f e r e n c e s are [ 3 H [ 1 2 ] , [ 1 7 ] ° Monna o b t a i n s i n t e r e s t i n g r e s u l t s when t h e norm range of t h e n o n - A r c h i m e d e a n normed l i n e a r s p a c e i s assumed t o have at most one a c c u m u l a t i o n p o i n t : 0. We s h a l l r e t a i n t h i s a s s u m p t i o n . In most of h i s work, Monna r e q u i r e s t h a t t h e v a l u a t i o n of t h e f i e l d of s c a l a r s be n o n - t r i v i a l ; t h i s , of c o u r s e , i s i m p o s s i b l e i n t h e c a s e of a v a l u e d s p a c e . D e f i n i t i o n 1 . 1 . A V - s p a c e X i s a s t r o n g l y p s e u d o - v a l u e d or a s t r o n g l y v a l u e d space which i s c o m p l e t e i n i t s norm t o p o l o g y and f o r w h i c h t h e r e e x i s t s a s e t of i n t e g e r s t»(X) and a r e a l number p > 1, s u c h t h a t ( 2 . 1 ) Q ( X ) = {0} U ( p " n : n € t»(X) } . D e f i n i t i o n 1 . 2 . A d i s c r e t e V - s p a c e i s a V - s p a c e such t h a t t h e s e t m(X) of D e f i n i t i o n 1 . 1 s a t i s f i e s ( 2 . 2 ) sup OD(X) = M f o r some M < ® . The t o p o l o g y of a d i s c r e t e V - s p a c e i s d i s c r e t e . A V - s p a c e such t h a t ( 2 . 3 ) SUP (j)(X) = eo 31 has a p r o p e r sequence c o n v e r g e n t t o 0 and hence i t s t o p o l o g y i s not d i s c r e t e . E x a m p l e s . In t h e space X d e f i n e d on pages 7, 8, t h e s e t of f o r m a l s e r i e s f o r w h i c h t h e s e t ( r , r ^ , •••} i s a s e t of i n t e g e r s i s a V - s p a c e s a t i s f y i n g ( 2 . 3 ) . Many o t h e r examples are c o n s t r u c t e d i n C h a p t e r 3° Convent i o n s . ( i ) In a l l of t h i s work, t h e symbol npn w i l l r e -t a i n t h e meaning a t t a c h e d t o i t i n D e f i n i t i o n 1=1. ( i i ) In t h e s e q u e l , whenever two or more V - s p a c e s w i l l be con-s i d e r e d s i m u l t a n e o u s l y , i t w i l l be assumed t h a t t h e v a l u e of p i s t h e same f o r a l l of t h e s e s p a c e s . Remark: A normed l i n e a r space c a n n o t be c o m p l e t e i f i t s f i e l d of s c a l a r s i s not c o m p l e t e w i t h r e s p e c t t o i t s v a l u a t i o n . F o r t h i s r e a s o n t h e r e i s no c o m p l e t e normed l i n e a r space o v e r t h e f i e l d R of t h e r a t i o n a l numbers w i t h t h e u s u a l v a l u a t i o n on R . In o o t h e c a s e of a V - s p a c e , any f i e l d F ( o f c h a r a c t e r i s t i c 0 — i n p a r t i c u l a r R Q i t s e l f ) i s a c c e p t a b l e s i n c e t h e v a l u a t i o n of F i s t r i v i a l . U nder t h e t r i v i a l v a l u a t i o n , -any f i e l d i s c o m p l e t e . The d e f i n i t i o n s , theorems and remarks of t h e f o l l o w i n g s e c -t i o n s of t h i s c h a p t e r , e x c e p t Th. 2 . 4 ( i i ) , do not depend on t h e c o m p l e t e n e s s of t h e V - s p a c e . They remain v a l i d f o r any space which s a t i s f i e s D e f i n i t i o n 1.1 e x c e p t f o r t h e c o m p l e t e n e s s r e q u i r e m e n t . P a r t ( i i ) of Theorem 2 . 4 r e q u i r e s c o m p l e t e n e s s . 32 2-2 E x i s t e n c e of d i s t i n g u i s h e d bases In t h i s s e c t i o n we prove that a V-space admits a d i s t i n g u i s h e d b a s i s (Theorem 2 . 2 ) . The proof of t h i s statement i s analogous, i n p a r t , to the proof of the ex i s t e n c e of an orthonormal b a s i s i n a H i l b e r t space ( [ 7 ] , P° 2 5 2 ; see a l s o [ 3 6 ] , p. 117). It i s made p o s s i b l e by the f o l l o w i n g improvement over Riesz's Lemma (Theorem 1 - 6 . l ) . Lemma 2.1. Let Y be a proper, c l o s e d subspace of a V-space X. There e x i s t s z £ X such that • |y - f o r a 1 1 y € Y -Proof : Let a s a t i s f y p " 1 < a < 1. Then | x | > a| z | i m p l i e s 1 x | > | z | f o r any p a i r x, z g X. By Theorem' 1-6.1, there e x i s t s z £ X\Y, such that Jy - z | > a| z |, f o r a l l y £ Y. Thus fly - z | > |z | f o r a l l y £ Y. Theorem 2 . 2 . A V-space admits a d i s t i n g u i s h e d b a s i s . Proof : Let D be the f a m i l y of a l l d i s t i n g u i s h e d subsets of a V-space X. D i s not empty since a s i n g l e point with non-zero norm forms a d i s t i n g u i s h e d subset of X. Let D be ordered by set i n c l u s i o n . It i s easy to see that a l i n e a r l y ordered sub-f a m i l y of D s a t i s f i e s the c o n d i t i o n s of Zorn's Lemma ( [ 3 6 ] , PP• 3 9 - 4 0 ; [18], p. 3 3 ) ° Therefore D contains at l e a s t one 33 maximal element H. We s h a l l show that [H] = X (see pp. 2 3 , 25 ). Suppose the c o n t r a r y . Then by Lemma 2 . 1 there e x i s t s z £ X \ [ H ] such that |y ~ z | > | z | f o r a l l y € [H] . a) If f o r each y £ (H), | y | * | z |, then |y + z j = Max {|y |, |zj} f o r a l l y € (H). b) If a) f a i l s , then f o r each y g (H) such that | y | = | z §, we have - | y € [H] f o r a l l a, 0 € F, a * 0 , 0 * 0 , and by the above i n e q u a l i t y : | ay + 0 Z | = § - |y - z | = | z | = Max ( l y | , | z f } . From a) and b) i t fo l l o w s that H IJ [zj i s a d i s t i n g u i s h e d subset of X, c o n t r a d i c t i n g the maxima-lity of H. Hence [H] = X and H i s a d i s t i n g u i s h e d b a s i s of X. The same argument a p p l i e s as usual to y i e l d the f o l l o w i n g : C o r o l l a r y 2 . 3 ° A V-space admits a d i s t i n g u i s h e d b a s i s which contains any given d i s t i n g u i s h e d s e t . In a Banach space B a complete b a s i s i s a sequence {b^} such that f o r every b g B there e x i s t s a unique sequence of s c a l a r s [a ] such that b = V a b . The c l a s s i c a l Paley-Wiener n n n n theorem ( [ 1 ] , [ 2 7 ] , [ 3 0 ] ) a s s e r t s that every sequence i n B, which i s " ' s u f f i c i e n t l y c l o s e " to a complete b a s i s , i s i t s e l f a b a s i s . 34 Arsove [ l ] has extended the Paley-Wiener theorem to a r b i t r a r y complete metric l i n e a r spaces over the r e a l or complex f i e l d with the usual v a l u a t i o n . Theorem 2 . 4 ( i i ) i s the V-space analogue of Arsove's Theorem 1 ( [ l ] , p. 3 6 6 ) . We do not r e q u i r e that H be countable. Theorem 2.4» Let H be a d i s t i n g u i s h e d subset of a V-space X and f be a mapping of H i n t o X such t h a t , f o r each h f H, ( 2 . 4 ) |h - a h f ( h ) | < | h | f o r some s c a l a r f 0 . Then ( i ) f(H) i s a d i s t i n g u i s h e d set of X; ( i i ) f (H) i s a d i s t i n g u i s h e d b a s i s i f H i s a d i s t i n g u i s h e d b a s i s . Proof : ( i ) | f ( h ) | = | h | f o r a l l h £ H. Let {h, : i = 1 ,2,••• fn} be a subset of H such that j f (h^)fl = r f o r i = 1, 2, n and some r > 0 . Let [|3^  : i = 1, 2, ° ° ", n} be any set of non-zero s c a l a r s . Then, from ( 2 . 4 ) , |h^ [ = | f ( h ^ ) | = r and 1(^7 h l h 2 + + £ h n ) -1 2 n i=1 1 n p Since H i s d i s t i n g u i s h e d , | V ~ h , | = r . It f o l l o w s that n i = l i»l This proves that f(H) i s a d i s t i n g u i s h e d set (see p. 2 5 ) . / 35 ( i i ) The proof i s a rewording of Arsove [ 1 ] , page . 3 6 7 ? i n which «yn«* and "A.1* must be r e p l a c e d by "a^ f ( h ^ ) " and " p - 1 " respec-n t i v e l y . Note that the proof r e q u i r e s the completeness of the space (see page 3 l ) . For examples and a p p l i c a t i o n s , see S e c t i o n 4 of Chapter 3 and Theorem 4 - 6 . 5 • 2-3 D i s t i n g u i s h e d f a m i l i e s of subsets The n o t i o n of d i s t i n g u i s h a b i 1 i t y was i n t r o d u c e d f o r subsets of a (pseudo-) valued space. It w i l l now be extended to f a m i l i e s of subset s. In a l l of t h i s s e c t i o n X i s a V-space. A set w i l l be c a l l e d t r i v i a l i f i t i s a subset of [ 9 ] , i . e . i f a l l i t s elements have norms equal to zero. D e f i n i t i o n 3 -1 • A f a m i l y { A a l of subsets of X i s a d i s t i n g u i s h e d  f a m i l y of subsets of X i f ( i ) A a n A a i s t r i v i a l f o r ± ( i i ) every non-empty subset B of |j A such that a a a ) | X I ^ 0 f o r each x £ B, b) no two elements of B belong to the same A a, i s a. d i s t i n g u i s h e d subset of X. C l e a r l y , a t r i v i a l set and any other subset of X form a d i s t i n g u i s h e d f a m i l y . A l s o , i f (A a} i s a d i s t i n g u i s h e d f a m i l y , 36 {_$a} i s a d i s t i n g u i s h e d f a m i l y of subsets i f B a C A a f o r a l l a. The f o l l o w i n g theorem gi v e s a c h a r a c t e r i z a t i o n of d i s t i n -guished f a m i l i e s of n o n - t r i v i a l subspaces of X. Theorem 3 . 2 . A f a m i l y {X a J of n o n - t r i v i a l ( c l o s e d or open) subspaces of X i s a d i s t i n g u i s h e d f a m i l y of subsets of X i f and on ly i f : ( i ) X a f)-X a i s t r i v i a l f o r t a_,, ( i i ) any union of d i s t i n g u i s h e d subsets of some or a l l of the X^'s i s a d i s t i n g u i s h e d subset of X. Proof : The s u f f i c i e n c y i s obvious. To prove the n e c e s s i t y , l e t B = U B where B i s not empty and B a i s e i t h e r empty or a d i s -ci t i n g u i s h e d subset of X^. Consider a f i n i t e l i n e a r combination of elements of B : n P • x = I I a i : x i i ' i = l j-1 where no ^ i s equal to 0 and where f o r each i £ [ l , 2, n], x.. € B a f o r j = 1, 2 , •••, p. J i Def i ne E P. , . a, x, ,. Then x, € B„ . j=1 i j i J i > - a.. Sin c e the B f l 's are d i s t i n g u i s h e d sets arid i [x^, x_,, *'", xn1 i s by ( i i ) of D e f i n i t i o n 3-1 a d i s t i n g u i s h e d set : n |x| = I V x | = Max |x. | = Max ( Max |x |) . K i < n 1 l<i<nU<j<p, J J Th i s shows that B i s a d i s t i n g u i s h e d s e t . 37 If the f a m i l y of subspaces co n s i d e r e d in Theorem 3 • 2 i s a f i n i t e f a m i l y of c l o s e d subspaces, the necessary and s u f f i c i e n t c o n d i t i o n of t h i s theorem can be c o n s i d e r a b l y weakened. Theorem 3 . 3 . A f i n i t e f a m i l y ^ = l ^ , x 2 , xn1 of non-t r i v i a l , c l o s e d subspaces of a V-space X forms a d i s t i n g u i s h e d f a m i l y of subspaces of X i f and only i f there e x i s t s _a f a m i l y « { H 1 # H 2 , — ' , H n} such t h a t : ( i ) B_ i s a d i s t i n g u i s h e d b a s i s of X^; i = 1, 2, n; ( i i ) 0 Hj i s empty f o r i j= j ; . n ( i i i ) H = U H, i s a d i s t i n g u i s h e d b a s i s f o r the c l o s e d sub-0 i = l 1 space X = [X, U X n U ° • ° U X 1. o u 1 2 n J Proof : N e c e s s i t y . For each i , X^ i s a V-space and admits a d i s t i n g u i s h e d b a s i s , . By ( i ) of D e f i n i t i o n 2 . 5 , the assump-t i o n that ^ i s a d i s t i n g u i s h e d f a m i l y i m p l i e s that X^ 0 X i s t r i v i a l f o r i * j . Since d i s t i n g u i s h e d bases do not con t a i n any t r i v i a l element, ( i i ) i s s a t i s f i e d . By ( i i ) of Theorem 3 ° 2 , H o i s a d i s t i n g u i s h e d s e t . C l e a r l y [H ] = X . S u f f i c i e n c y . It i s easy to see that X^ D X^ i s t r i v i a l f o r i * j . We must show that ( i i ) of D e f i n i t i o n 3-1 i s s a t i s f i e d . Let [x^, X£, 0 0 0 , x m l , m S n , D e such that J x ^ | * 0 and assume that the X,*s are reindexed i n such a way that x. P X, , 1 - 1 1 f o r i = 1, 2 , ••», m. For each f i x e d i , 1 < i < m, there e x i s t s a n o n - i n c r e a s i n g expansion of x, i n terms of H, : 38 x. = ) a. .y. ,, • a, . f- 0. 1 L i J i j ' u J>1 ' • Ac c o r d i n g to Lemma l - 7 « 5 |x. | = sup |y. . |. Suppose that | y i k f i = sup | y i 1 | f o r k <, p., j |y i k I < sup | y ± j | f o r k > p i < p^ i s n e c e s s a r i l y f i n i t e . C onsider now any set of s c a l a r s {3^, 8 Pml» We c a n assume without l o s s of g e n e r a l i t y that B ^ 0 f o r each i < m. , , P . x, . If we can show that i = l i i ( 2 . 5 ) | x | = Max f x |, l<i<m we w i l l have proved that ^ i s a d i s t i n g u i s h e d f a m i l y of subsets of X. Since H q i s a d i s t i n g u i s h e d set by assumption, we have: n p.. n oo (2.6) |>|- | I 1 h a i j 7 i i . J I P ^ . y ^ i , i = l j = l i - 1 j = P . + l n P i ( 2 . 7 ) | Y J 3 a y | = Max |y. | = Max fj x |, U /- i U i J - i < i < n X J K i < n n on l ( 2 . 8 ) | J y 3 a y j < Max |y | = Max f x J. The e q u a l i t y ( 2 . 7 ) i s guaranteed by the f a c t that no canc/ella-Pi. t i o n of terms can a r i s e i n the f i n i t e sum ) t _•, since/ the i - i Z j - i 39 H^'s are assumed to be d i s j o i n t . From ( 2 . 6 ) , ( 2 . 7 ) and ( 2 . 8 ) , i t f o l l o w s that ( 2 . 5 ) i s t r u e , and the proof of s u f f i c i e n c y i s completed. If a f i n i t e f a m i l y {X^, X^, X^} of n o n - t r i v i a l , c l o s e d subspaces of a V-space i s a d i s t i n g u i s h e d f a m i l y , the c l o s e d sub space [X^ U Xg U °°° U x n ] w i l l be c a l l e d the d i r e c t sum of X^, X^, X^. This d i r e c t sum w i l l be denoted by © © ••• ® . Conversely, whenever i n the sequel the symbol © w i l l be used, i t w i l l be understood that the subspace i n v o l v e d form a d i s t i n g u i s h e d f a m i l y of c l o s e d subspaces. (Com-pare with [ 7 ] , PP. 3 7 , 2 5 6 ) . The f o l l o w i n g c o r o l l a r y to Theorem 3 . 3 i s e a s i l y proved: C o r o l l a r y 3 • 4» The decomposition of any poin t of X, © X„ © ••• © X as a sum of element s of the X,'s i s unique, 1 2 n 1 except f o r order and a d d i t i o n of t r i v i a l elements. In t h i s s e c t i o n we have e x h i b i t e d some analogy between H i l b e r t spaces and V-spaces. The analogy i s a consequence of th s i m i l a r i t y of the r o l e s p l a y e d by the concepts of o r t h o g o n a l i t y and d i s t i n g u i s h a b i l i t y i n the two types of spaces. The d e f i n i -t i o n s and theorems of t h i s s e c t i o n should be compared with the d e f i n i t i o n s and theorems on complete orthonormal bases, ortho-gonal complements, e t c . , i n the theory of H i l b e r t spaces; [ 7 ] , [ 1 4 ] , [ 1 9 ] , [ 3 6 ] . In the next s e c t i o n an important d i f f e r e n c e between the two s t r u c t u r e s w i l l become apparent. 40 2-4 D i s t i n g u i s h e d complements In a H i l b e r t space, the orthogonal complement of a set A i s d e f i n e d to be the set of a l l the elements of the space which are orthogonal t o a l l the elements of A. In a V-space, we introduce d a c o r r e s p o n d i n g n o t i o n : The d i s t i n g u i s h e d adjunct A of a sub-set A of a V-space X i s d e f i n e d by A d = [x £ X : {x} and A form a d i s t i n g u i s h e d p a i r of subsets of X}. In Theorem 4-l# simple p r o p e r t i e s of d i s t i n g u i s h e d adjuncts are s t a t e d . ( i i ) expresses the f a c t that the d i s t i n g u i s h e d adjunct of a set A i s the largest, set which forms with A a d i s t i n g u i s h e d p a i r of subsets of X. Parts ( i ) , ( i i i ) , ( i v ) and (v) should be compared with Theorems 1, 2, 3 , 4 of [ 1 4 ], P« 24» The proofs of the statements f o l l o w d i r e c t l y from o u r . d e f i n i t i o n s and are omitted. Theorem 4 • 1 • If A and B are "subsets of a V-space, then each of the f o l l o w i n g statements i s v a l i d : ( i ) A fl A d i s - t r i v i a l , ( i i ) If (A, B) i s a d i s t i n g u i s h e d f a m i l y of subsets of X, then A c B d and B e A d. ( i i i ) A c A d d . ( i v ) If A c B, then B d tz A d . (v) A d = A d d d . In a H i l b e r t space the orthogonal complement of any set i s a c l o s e d subspace ( [ 1 4 ] , P« 2 4 , Th. 2 . 6 ) . However, the same i s 41 not true of the d i s t i n g u i s h e d adjunct of a set i n a V-space. The f o l l o w i n g example i l l u s t r a t e s t h i s f a c t . Let A = S(9, r) and flz| > r; then- z £ A and f o r a l l y € A, y + z £ A ; c l e a r l y d d y => (y + z) - z does not belong to A . Thus, A i s not, i n g e n e r a l , a subspace of the V-space, even when A i t s e l f i s a subspace. Two consequences of the discrepancy j u s t mentioned are, f i r s t , that the n o t i o n of d i s t i n g u i s h e d adjunct w i l l not be u s e f u l i n the sequel; and, secondly, that the non-uniqueness of the d i s t i n g u i s h e d complement of a c l o s e d subspace (as d e s c r i b e d i n the f o l l o w i n g d e f i n i t i o n ) w i l l have a r o l e i n the theory. D e f i n i t i on 4»2. Two c l o s e d subspaces Y , Z , of a V-space X are s a i d to be d i s t i n g u i s h e d complements of one another i f ( i ) Y and Z f orm a d i s t i n g u i s h e d p a i r of subsets of X, ( i i ) Y € > Z = X. It i s c l e a r that the only d i s t i n g u i s h e d complement of [9] i s X, and c o n v e r s e l y . If Y i s a n o n - t r i v i a l , c l o s e d , proper sub-space of X, Y admits a d i s t i n g u i s h e d complement, but, i n general, i t i s not unique. This i s expressed i n Theorem 4 - 4 , i n which we use the f o l l o w i n g t erminology. D e f i n i t i o n 4«3 « Let Y and Z be c l o s e d subspaces of a V-space, with d i s t i n g u i s h e d bases H ( Y ) and H ( z ) r e s p e c t i v e l y . H ( z ) i s c a l l e d an extension of H ( Y ) to Z i f H . ( Y ) C . H ( Z ) . (This i m p l i e s Y c Z ) . 42 Theorem 4»4« Let Y be a n o n - t r i v i a l , c l o s e d , proper subspace of a V-space X. Let H ( Y ) be"any d i s t i n g u i s h e d b a s i s of Y and H be any extension of H ( Y ) to X. The subspace [ H \ H ( Y ) ] i s a non-t r i v i a l , c l o s e d , proper subspace of X which i s a d i s t i n g u i s h e d complement of Y . The theorem i s e a s i l y proved, u s i n g C o r o l l a r y 2 . 3 and Theo-rem 3-3« It does not s t a t e that two d i f f e r e n t p a i r s ( H ^ ( Y ) , H ^ ) and (H^CY), E^) n e c e s s a r i l y w i l l generate d i s t i n c t d i s t i n g u i s h e d complements of Y . As a simple example, l e t X have a d i s t i n g u i s h e d b a s i s formed by three elements x^, x^, x^ with j x ^ | = I x 2 I = I ^ l " ^et Y = [x^] and H ( Y ) = i x ^ j . Three p o s s i b l e extensions of H ( Y ) to X are : H 1 = {xlf x 2, x^}, H 2 = |x 1, x1 + x 2 , x^}, H 3 = {x 1 # x 2 + x^, x^}. The d i s t i n g u i s h e d complements of Y generated by the p a i r s ( H ( Y ) , B ^ ) and ( H ( Y ) , H ) are both equal to [ x 2 , x ] but that generated by ( H ( Y ) , E^) d i f f e r s from [x,>, ] . 2-5 Notes ( i ) The concept of d i s t i n g u i s h a b i l i t y has been i n t r o d u c e d by Monna under a d i f f e r e n t name and through another formal d e f i n i -t i o n . In h i s e a r l y papers, [ 2 4 ] , Monna uses the term "pseudo-orthogonal"; i n h i s l a t e r work [ 2 5 ] , he uses the word "orthogo-n a l " . In a s t r o n g l y non-Archimedean normed l i n e a r space, a 43 p o i n t x i s s a i d t o be orthogonal to a point y i f the d i s t a n c e from x t o the l i n e a r subspace (y) i s equal to the norm of x. ( [ 2 4 ] , V, p. 197; [ 2 5 ] , I, P. 480). It i s e a s i l y v e r i f i e d that y i s then orthogonal to x. The equivalence of t h i s d e f i n i t i o n of o r t h o g o n a l i t y and of our d e f i n i t i o n of d i s t i n g u i s h a b i l i t y i s i n d i c a t e d i n the f o l l o w -i n g theorem. ( c f . Monna, [ 2 4 ] ) : Theorem 5•1» Let A be a subset of a V-space X. For x £ A, l e t A^ denote the l i n e a r subspace (A%{x])« Then, A i s a d i E x -t i n g u i s h e d subset of X i f and only i f , f o r a l l x £ A: x £ [9] and d i s t a n c e (x, A ) = |x|. The proof i s omitted. From the notion of o r t h o g o n a l i t y , Monna c o n s t r u c t s a theory of orthogonal sets and orthogonal complements q u i t e analogous to our theory of d i s t i n g u i s h e d sets and complements. In [ 2 5 ] , i t i s assumed that the v a l u a t i o n of the f i e l d of s c a l a r s i s not t r i v i a l . Use i s not made very e x t e n s i v e l y of "complete orthogonal** ( d i s t i n g u i s h e d ) bases, which e x i s t only under s p e c i a l assumptions, such as l o c a l compactness and separa-b i l i t y . ' -An important t o o l used by Monna i s the concept of a " p r o j e c -tion"'. We have postponed the i n t r o d u c t i o n of p r o j e c t i o n s i n our theory u n t i l l i n e a r operators are s t u d i e d . ( i i ) I n g l e t o n ( [ 1 7 ] , p. 4 2 ; see al s o [ 2 5 ] , I, p. 475) d e f i n e s a 44 s p h e r i c a l l y complete t o t a l l y non-Archimedean metric space ( f i e l d ) as a t o t a l l y non-Archimedean m e t r i c space ( f i e l d ) i n which every f a m i l y of c l o s e d b a l l s l i n e a r l y ordered by i n c l u -s ion has non-void i n t e r s e c t i o n . S p h e r i c a l completeness i m p l i e s completeness ( [ 2 5 ] , I, P- 4 7 6 ) . In g e n e r a l , com-pl e t e n e s s does not imply s p h e r i c a l completeness, but, i f the norm ( v a l u a t i o n ) s a t i s f i e s ( 2 . 1 ) and ( 2 . 3 ) of D e f i n i t i o n 1 .1 , then completeness i m p l i e s s p h e r i c a l comp l e t en e ss ( [ 25 ] , H , P« 4 8 6 ) . T h e r e f o r e , a V-space i s s p h e r i c a l l y complete. Monna ( [ 2 5 ] , H I , P« 466) has shown that the e x i s t e n c e of a complete orthogonal ( d i s t i n g u i s h e d ) b a s i s i n a non-Archimedean normed l i n e a r space i s r e l a t e d to the completeness of the space and the s p h e r i c a l completeness of the f i e l d of s c a l a r s , when the v a l u a t i o n of the f i e l d i s n o n - t r i v i a l . It i s p o s s i b l e that a r e f o r m u l a t i o n of the arguments of Monna to the case of a f i e l d of s c a l a r s p r o v i d e d with a t r i v i a l v a l u a t i o n c o u l d l e a d to a proof of e x i s t e n c e of a d i s t i n g u i s h e d b a s i s f o r a V-space. Our proof (Theorem 2 . 2 ) i s more d i r e c t and shows that completeness c o n d i t i o n s are unnecessary (see page 3 1 ) . 2-6 V-Algebras In t h i s s e c t i o n we s h a l l c o n s i d e r V-spaces X on which a m u l t i p l i c a t i o n i s d e f i n e d , i . e . such that to each p a i r (x, y) £ X x X there corresponds a unique "product" xy £ X. Remark: In t h i s S e c t i o n we s h a l l d e f i n e some elements of a 45 V-space by use of sequences and s e r i e s . Since a V-space can be a pseudo-normed space, the l i m i t of a sequence or the sum of a s e r i e s are not n e c e s s a r i l y unique. For t h i s reason we make the f o l l o w i n g n o t a t i o n a l convention: Convent i on. In the sequel, the r e l a t i o n M x = y n means that |x - y | = 0 ; s t r i c t i d e n t i t y between x a n d y i s i n d i c a t e d by the symbol r*x s y'*. The d e f i n i t i o n s and theorems of t h i s S e c t i o n are simple m o d i f i c a t i o n s of the d e f i n i t i o n s and theorems of the c l a s s i c a l theory of normed r i n g s ( [ 2 2 ] , [ 2 6 ] ) . D e f i n i t i on 6 . 1 . A V-space X with a m u l t i p l i c a t i o n i s c a l l e d a V- a l g e b r a i f f o r a l l x, y ^ X and a l l s c a l a r s a: ( 2 . 9 ) a(xy) s (ax)y E x ( a y ) , ( 2 . 1 0 ) x(yz) B (xy)z, ( 2 . 1 1 ) x(y + z) s xy + xz, (y + z)x s yx + zx, ( 2 . 1 2 ) |xy | < |x|-|y |. We a l s o assume the e x i s t e n c e of an i d e n t i t y , i . e . of an element e such that ( 2 . 1 3 ) xe s e x E X f o r a l l x f X r ( 2 . 1 4 ) l e i = 1. 46 X i s s a i d t o be a commutative V-alqebra i f ( 2 . 1 5 ) xy E yx f o r a l l x, y g X. C o n d i t i o n ( 2 . 1 2 ) i m p l i e s the c o n t i n u i t y of.the m u l t i p l i c a -t i o n , i n both v a r i a b l e s . Thus, i f x -» x and y -* y, ' ' n n ' n = 1, 2 , then x y _ xy . ' ' ' n n If a V-space X has a l l the p r o p e r t i e s of a V-algebra except f o r the e x i s t e n c e of an i d e n t i t y , the i d e n t i t y e can be f o r m a l l y a d j o i n e d . The c l a s s i c a l process of adjunction of an i d e n t i t y to an algebra i s d e s c r i b e d i n [ 2 2 ] , p. 59 and i n [.26] , p. 157- Con-d i t i o n ( 2 . 1 4 ) i s not e s s e n t i a l but i s made to s i m p l i f y the proofs of c e r t a i n statements? From ( 2 . 1 2 ) , | e | > 1 i n a l l cases. There-f o r e , d i v i d i n g a l l norms by | e | w i l l not change the topology and w i l l give to the i d e n t i t y a new norm s a t i s f y i n g ( 2 . 1 4 ) . As u s u a l , we denote by x 1 1 the product xx--»x of n elements equal t o x. x° 5 e f o r a l l x £ X. D e f i n i t i o n 6 . 2 . An element e T w i l l be c a l l e d a p s e u d o - i d e n t i t y i f e 1 = e . D e f i n i t i o n 6 . 3 . Let x be an element of a V- a l g e b r a X. ( i ) x i s s a i d t o be pseudo-regular i f there e x i s t s an element x ^ such that -1 -1 xx = x x = e. x ^ i s c a l l e d a pseudo-inverse of x. ( i i ) x i s s a i d to be r e g u l a r i f there e x i s t s an element x ^ such that 47 xx = x x s e. x ^ i s c a l l e d the i n v e r s e of x. ( i t can be proved that such an element i s unique). ( i i i ) If x i s not (pseudo-)" r e g u l a r i t i s s a i d to be s i n g u l a r . No element of [9] i s (pseudo-) r e g u l a r f o r otherwise 1 = | xx" 1 | < | x | J x _ 1 1 = 0 . Theorem 6«4« Let x 1 be a pseudo-inverse of an element x of a V-alg e b r a X. Then y i s a pseudo-inverse of x i f and only i f y = x 1 . Consequently, any two pseudo-inverses of x have equal norms. Proof: The s u f f i c i e n c y i s obvious. The n e c e s s i t y f o l l o w s from ( 2 . 1 0 ) - ( 2 . 1 2 ) and the f a c t that f o r some 9', 9" ^ [ 6 ] • ( x - 1 — y ) x «= 9'; J U " 1 - y ) x x - 1 | = | 0 » x _ 1 | = 0 ; | ( x _ 1 - y ) x x - 1 | = I U " 1 - y ) ( e + 9«) [ = | x _ 1 - y|. Lemma 6 .5» Let X be a V-algebra, x £ X and |x| < 1. Then CO ( i ) the sums of the s e r i e s ^ x n are pseudo-inverses of / \ n=0 (e - x) ; ( i i ) f o r every pseudo-inverse x' of (e - x ) : x + (e - x') = x(e - x ? ) , x + (e - x 1 ) = (e - x')x. j The proof i s obtained by d i r e c t v e r i f i c a t i o n . (See [ 2 2 ] , pp. 6 4 - 6 6 ) . 48 Theorem 6 . 6 . Let Y denote the set of pseudo-regular p o i n t s X. If y f Y, x £ X and |x - y | < | y - 1 |~ 1, then ( i ) x € Y; ( i i ) |x I - |y X | ; ( i i i ) Ix"1 - y " 1 ! < ly^ P j x - y | < l y " 1 ! ; ( i v ) Y i s an open subset of X and the mapping y -* y d e f i n e d on Y, i s continuous. Proof: ( i ) x s (x - y) + y = y[y _ 1(x - y) + e] Si n c e |y '"(x - y),| < 1, i t f o l l o w s from Lemma 6 . $ that z = y ( x - y ; + e = y x has a pseudo-inverse z 1 . Thus, x has a pseudo-inverse -1 -1 -1 x = z y ( i i ) From Lemma 6 . 5 , CO x_1 = ^ £ f.~y - 1 ( x ~ y ) l n ) y - 1 = y - 1 + • • • n=0 Since I[y_1(x - y ) j n + 1 | < l i y - ^ x - y ) } n | , n - 0, 1, 2 , I* I = |y |. ( i i i ) The c o n c l u s i o n f o l l o w s from the e q u a l i t y oo x _ 1 - y _ 1 = ( £ {-y-1(x - y ) } n ) y _ 1 n = l = -y "*-(x - y)y 1 + ( i v ) i s a consequence of ( i i i ) . I 49 Definition 6 . 7 - The spectrum (j-(x) of an element x of a V-algebra X i s the set of s c a l a r s A. f o r which (x - A,e ) i s s i n g u l a r . Theorem 6 * 8 . Let x £ X, 0 < JxJ < 1. If f o r some s c a l a r (i, | x - [ie \ < 1, then ( i ) a(x) i s empty or a ( x ) = {(J.]; ( i i ) f o r A. f" [i-f the pseudo-inverses of (x - A,e) are the sums of the s e r i e s (2.16) - Y (x - u e )n n + 1 ' n=0 ( A - ~ ^ and s a t i s f y ( 2 . 1 7 ) | ( x - A . e ) - 1 | = |x - A.e| = 1. P r o o f : We note f i r s t that t h i s theorem i s an extension of Lemma 6.5- Indeed, i f | x | < 1, we take |i = 0 . ( i ) i s a consequence of ( i i ) . ( i i ) Since J (x - H-e) n| < |x - p,e|n, the s e r i e s (2.16) converges, Using the c o n t i n u i t y of the m u l t i p l i c a t i o n , we v e r i f y d i r e c t l y that i f y i s a sum of (2.16), y(x - n,e) - (A. - [i)y = y ( x - Xe) = e, (x - [ie)y - (A. - [i,)y = (x - A.e)y = e. . ( 2 . 1 7 ) f o l l o w s from (2.16) and the f a c t that a V^-algebra i s a s t r o n g l y valued space. A d i r e c t proof of the f o l l o w i n g theorem i s s i m i l a r to that of Theorem 6.8. 50 Theorem 6 .9« Let x £ X, | x | > 1. If f o r some s c a l a r [i, (x - He) i s pseudo-regular and | (x - \ie) ^ | < 1, then ( i ) cr( x) i s empty; ( i i ) f o r X, * H, the pseudo-inverses of (x - A,e) are the sums of the s e r i e s oo n + 1 £ (A, - p,) n[(x - t i e ) " 1 ] 1 1 n=0 and s a t i s f y | (x - A . e ) - 1 | = | ( x - i x e ) - 1 ! . Remarks. ( i ) As i n the c l a s s i c a l theory of normed r i n g s , the s i n g u l a r i t y or r e g u l a r i t y of an element of a V-algebra depends on i t s b e l o n g i n g to some maximal i d e a l of the alg e b r a . (For the terminology, see [26], p. 159 or [ 7 ] , P- 3 8 . ) One can prove that a) any i d e a l of a V-algebra i s contained i n a maximal i d e a l ( [ 2 2 ] , p. 5 8 ; [26], p. 1 5 9 ) ; b) a maximal i d e a l i s c l o s e d ( [ 2 2 , p. 6 8 ] ; c) an element of a V-al g e b r a i s pseudo-regular i f and only i f i t does not belong to any maximal i d e a l of the algebra ( [ 2 2 , p. 64; [ 2 6 ] , p. 159)» It i s al s o easy to prove that i f Y .is an i d e a l then j e - y | > 1 f o r a l l y $ Y. (Compare with [ 2 2 ] , Th. 22D, p. 6 8 ) . ( i i ) The theory of Banach algebras ( [ 2 2 ] , [26]) i s p a r t i a l l y based on the f a c t that any Banach f i e l d i s completely isomorphic to the f i e l d of the complex numbers (with i t s u s u a l t o p o l o g y ) . From t h i s r e s u l t one attempts to c h a r a c t e r i z e the maximal i d e a l s of the a l g e b r a . 51 There does not seem to be any i n t e r e s t i n g analogue of t h i s theory i n the case of V-algebras. 52 CHAPTER 3 EXAMPLES OF V-SPACES ASYMPTOTIC AND MOMENT SPACES 3-1 Int roduct i on C e r t a i n spaces of f u n c t i o n s , mapping a Hausdorff space i n t o a normed l i n e a r space, can be normed i n such a way that they be-come V-spaces. In t h i s chapter we s h a l l d e s c r i b e two methods to generate such V-spaces. In the f i r s t type of V-spaces, the norm c o n s i d e r e d w i l l c h a r a c t e r i z e the asymptotic behaviour of the f u n c t i o n s ; the r e s u l t i n g spaces w i l l be c a l l e d - "asymptotic spaces". In the second type, we s h a l l a s s o c i a t e with each f u n c t i o n a sequence of s c a l a r s , c a l l e d "moments", and the norm of a f u n c t i o n w i l l depend on the f i r s t non-zero moment; the r e s u l t i n g spaces w i l l be c a l l e d "moment spaces". 3-2 The 0 arid o r e l a t i o n s a) Let be a Hausdorff space and l e t P and S be a r b i t r a r y s e t s . We c o n s i d e r functions of the three v a r i a b l e s A. £ J^, p £ P and s £ S. The v a r i a b l e A. w i l l be c a l l e d the asymptotic v a r i a b l e ; p and s w i l l be c a l l e d the primary and secondary para-meters r e s p e c t i v e l y . b) Let A.Q be a f i x e d n o n - i s o l a t e d point of J^. The a b b r e v i a t i o n "cd-nbhd of A. " w i l l stand f o r " c l o s e d o neighbourhood of A, i n A, d e l e t e d of the point A, i t s e l f " . A 53 cd-nbhd of A, has non-void i n t e r i o r . A f i n i t e i n t e r s e c t i o n of o cd-nbhds of A. i s a cd-nbhd of A. . o o c) If f and g are two f u n c t i o n s d e f i n e d on V x P X S, where V i s some cd-nbhd of A,q, and with range i n a (pseudo-) normed space (which may be d i f f e r e n t f o r the two f u n c t i o n s ) , then the r e l a t i o n f = 0(g), A, \ Q , w i l l mean that there e x i s t , f o r each s £ S, a p o s i t i v e constant a(s) and a cd-nbhd V(s) of A. such that o II f (A., p, s)|| < a(s)||g(A,, p, s)|] ( 3 - D f o r a l l A. £ V(s) and a l l p £ P. In t h i s i n e q u a l i t y the norms are those of the ap p r o p r i a t e range spaces. S i m i l a r l y , the r e l a t i o n f = o (g), A, m A, O, w i l l mean that f o r any e > 0, there e x i s t s , f o r each s £ S, a cd-nbhd V ( s , £) of A, such that o Hf (A., p, s)|| < e||g(A., P, s)|| f o r a l l A, £ V(s, e) and a l l p £ P. In u s i n g the 0 and o symbols, the s p e c i f i c a t i o n A, — A. w i l l u s u a l l y be omitted. These 0 and o r e l a t i o n s have the f o l l o w i n g p r o p e r t i e s : 54 ( 3 - 3 ) 0 ( 0 ( f ) ) = 0 ( f ) , ( 3 . 4 ) 0 ( o ( f ) ) - o ( 0 ( f ) ) - o ( o ( f ) ) = o ( f ) , ( 3 - 5 ) G ( f ) + 0 ( f ) = 0 ( f ) + o ( f ) = 0 ( f ) , ( 3 . 6 ) o ( f ) + o ( f ) = o ( f ) , ( 3 . 7 ) 0 ( f ) - 0 ( g ) = 0 ( f g ) , ( 3 . 8 ) 0 ( f ) - o ( g ) = o ( f ) . o ( g ) = o ( f g ) . P r o p e r t i e s ( 3 - 7 ) and (3=8) apply when the range spaces are (pseudo-) normed r i n g s . The proofs are immediate and the formulae can be extended to combinations of any f i n i t e number of order symbols.. For those and other r e l a t i o n s , see [ 9 ] , Chapter 1. d) D e f i n i t i o n 2 . 1 . A sequence ff } of f u n c t i o n s i s c a l l e d an n asymptotiic sequence (as A, -» A.q) i f ( i ) f i s d e f i n e d on V x P X S, where V i s some n n n cd-nbhd of A, ; o ( i i ) a l l f have the same range space; and ( i i i ) f .1 = o(f ) f o r each n. n + l n 3~3 Asymptotic spaces: D e f i n i t i o n a) Let A , P, S be as i n S e c t i o n 2 and A* = A \ { ^ . 0 ] » Let B and B Q be two (pseudo-) normed l i n e a r spaces. The (pseudo-) norms on both spaces w i l l be denoted by || • |j . Let N be a set of i n t e g e r s such that sup N = ». The ordering-on N i s the n a t u r a l o r d e r i n g of the i n t e g e r s . For n € N, o"°( n) = n / o"^^n^ = 0"(n) denotes the successor of n i n N 55 i t h and (f (n), j = 2, 3» **• denotes the j successor of n i n N; the element m £ N such that & (m) = n i s denoted by Q- "*(n). b) D e f i n i t i o n 3.1. A f a m i l y of f u n c t i o n s $ = lcpn = ri £ N} i s c a l l e d an asymptotic s c a l e (as A, -. A,q) i f f o r each n £ N : ( i ) <pn i s d e f i n e d on A' X P X, S and have range i n B q ; and ( i i ) the sequence {m , : j = 0, 1, 2, •••) i s an asymptotic sequence. In analogy with the terminology of J . G. van der Corput [ 3 8 ] , [ 3 9 ] we use the f o l l o w i n g D e f i n i t i o n 3.2. A f u n c t i o n f d e f i n e d on V x P X S, where V i s some cd-nbhd of A,^ , and with range i n a (pseudo-) normed space i s s a i d to be a s y m p t o t i c a l l y f i n i t e with respect to an asymptotic  s c a l e $ = [cpn J n € NJ i f there e x i s t s n £ N such that f = 0(q> n). c) Let X be the l i n e a r space of a l l f u n c t i o n s d e f i n e d on A* X P X S, with range i n B, which are a s y m p t o t i c a l l y f i n i t e with respect to a given asymptotic s c a l e . For each x g X, d e f i n e ( 3 - 9 ) u)( x) = sup{n g N : x = 0(<pn)}, and, f o r some f i x e d r e a l p, 1 < p < <», (3.10) I ) = P - » ( X ) . The f u n c t i o n d e f i n e d on X by ( 3 - 9 ) and (3.10) w i l l be c a l l e d the ^-asymptotic norm on X. 56 Since the asymptotic behaviour of a f u n c t i o n , as X -» A,o, i s e n t i r e l y determined by i t s values on any set of the form V x P x S, where V i s a cd-nbhd of A, q, the ^-asymptotic norm of the d i f f e r e n c e of two f u n c t i o n s i n X which are ^ i d e n t i c a l on V x P X S i s equal to zero. Conversely, i f a f u n c t i o n i s de-f i n e d on V x P X S and i s a s y m p t o t i c a l l y f i n i t e with respect to then i t can be a r b i t r a r i l y extended to a l l of A' x P x S and the ^-asymptotic norm of the d i f f e r e n c e of any two of i t s exten-sions i s equal t o zero. d) Theorem: 3 »3• The space X, under the ^-asymptotic norm, i s a V-space. Proof; Using ( 3 ° 3 ) - ( 3 - 9 ) one e a s i l y v e r i f i e s t h a t , under the norm ( 3 . 1 0 ) , X has the p r o p e r t i e s of a s t r o n g l y pseudo-valued space ( D e f i n i t i o n s 1-1.1 and 1 - 1 . 2 ) . The ^-asymptotic norm s a t i s f i e s ( 2 . 1 ) of D e f i n i t i o n 2 -1 . 1 and a l s o ( 2 . 3 ) . To prove the completeness of X, c o n s i d e r an a r b i t r a r y Cauchy sequence {s^ : i = 0 , 1, 2, •••}. Let y = s , y. = s. - s. , f o r i = 1, 2, • • • o 0 1 1 1 - 1 ' ' From Theorems 1 -4 .1 and l - 4 » 2 , i t i s s u f f i c i e n t t o prove the convergence of a p a r t i c u l a r rearrangement of the s e r i e s ( 3 . 1 1 ) £ vt i=0 Without l o s s of g e n e r a l i t y , we can assume that none of the y^'s has zero-norm. Let 57 M = (n £ N : | y i | = p " f o r some i } , q = i n f M. Since [s^] i s a Cauchy sequence, i t f o l l o w s that q > A l s f o r each n £ M, the number of y ^ T s with norms equal to p n i s f i n i t e . Let x be t h e i r sum. It f o l l o w s that n ( 3 . 1 2 ) Ixnl- f o r a 1 1 n € N and that the s e r i e s ao (3-13) I x n n=q n^M can be c o n s i d e r e d as a rearrangement of ( 3 . H ) . We now f i x an a r b i t r a r y value s £ S f o r the secondary parameter. It f o l l o w s from ( 3 . 1 2 ) that f o r each n g M, there e x i s t constant a[n, s] > 0 and a cd-nbhd V[n, s] of "K such that ||x ( n ^ ( \ , p, s)|| < a[g.(n), s]||<po.(n) C^, P» s)JJ < ^ a [ n , s][|cpn p, s)|| f o r a l l A. £ V[g.(n), s] and f o r a l l p £ P. 58 We can assume without l o s s of g e n e r a l i t y that these V[n, s ] ' s are nested; and are s e l e c t e d i n such a way that t h e i r i n t e r s e c t i o n i s v o i d . * Then, f o r a l l j > 0, " V ( ) ( A " P ' S ^ ~ 2 _ J a [ n ' S ^ t ^ n ( A - ' P ' s ) " f o r a l l X. £ V [ a J ( n ) , s] and f o r a l l p £ P. We s h a l l now d e f i n e an element x of X by s p e c i f y i n g i t s values on X P X {s}. For A, £ J^1 \ V [ q , s] we de f i n e x(A,, p, s) = 0 f o r a l l p £ P. For A. £ V[q, s ] , there e x i s t s an i n t e g e r N (A., s ) : N (A., s) = Max [n f N : A. £ V[n, s] } If A. £ V[n, s] then N (A., s) > n. For A, g V[q, s] we d e f i n e x(A., p, s) = ^ x n (A,, p, s) f o r a l l p £ P. q<n<N(A., s) nfM *In a d d i t i o n to these requirements, the choice of the cd-nbhds V[n, s] i s guided by the c o n d i t i o n that f o r A- 6 ^Ltr (<0, s ] , j - 1, 2, ••• |). j (*.,P,s)|| < | ^ f ] 1 ™ . ' ^ (A.,p,s)/} V U ) a [ ^ ( q ) , s ] V X ( q ) f o r a l l p 6 p. which i m p l i e s ||T . (A.,P,s)|l < 2" J U l q ' 8 1 fU (A., p, s)fl V U ) a [ ^ ( q ) , s ] T ^ f o r a n p € P o 59 We s h a l l now show that x i s a l i m i t of the sequence ( 3 • 1 3 ) and thus of ( 3 . 1 1 ) . Let £ > 0 be giv e n ; there e x i s t s J such that f o r a l l j > J, p~ J < £. We a s s e r t t h a t , f o r a l l j > J , j £ M, q<n<j • n^M or, e q u i v a l e n t l y , that (x - £ x n ) = 0 ( ^ j ) . q<n< j ngM Indeed, f o r each s £ S, f o r A, £ V [ j , s ] , (j £ M), A. £ V[o,(j), s ] , A. £ V [ o . 2 ( j ) , s ] , A, £ V[N(A,, s ) , s] and J|x(A., p, s) - £ x n ^ ' P ' n^M = |j x ^ ^ (A., p, s) + x 2 (A.; p, s) + •*• + x N ^ s)(A.,p,s)f| < 2 a [ j , s]||q>.,(?C, p, s)||, f o r a l l p £ P. This completes the p r o o f . The above proof i s modelled a f t e r the second part of the proof of [ 3 3 ] , Theorem 1 ( a l s o [ 3 2 ] , Th. 4 . 2 ) . In [ 3 3 ] , the range space B i s a Banach space; we have shown that the complete-ness of X does not r e q u i r e the completeness of B. If B i s a (pseudo-) normed r i n g , the product xy of two f u n c t i o n s x, y £ X i s d e f i n e d by xy(A,,p,s) = x (A,, p, s) «y (A,, p, s ) f o r a l l A,, p, s. 60 If B q i s a (pseudo)-normed r i n g , a s i m i l a r d e f i n i t i o n of the product of two elements of § can be g i v e n . Theorem 3»4- Let B and B q be (pseudo-) normed r i n g s . If f o r a l l m, n £ N, cpmcpn = 0 ( ( p m + n ) , then x s a t i s f i e s the p r o p e r t i e s ( 2 . 9 ) , (2.10) and (2.11) of a V - a l g e b r a . P r o o f : If x, y £ X and | x | = p _ m , | y | = p~ n, then by ( 3 - 7 ) and ( 3 - 3 ) : Hence, | xy \ < 0 ( m + n ) = |x|°|y| and (2.1l) i s s a t i s f i e d . One v e r i f i e s e a s i l y that (2 . 9 ) and (2.10) h o l d . e) Let x £ X have, i n the ^-asymptotic norm on X, the f o l l o w i n g expansion: (3.15) x = a Q X O + a 1 x 1 + a 2 x 2 + ••• , x± £ X, CL £ F. The expansion (3»15) i s s a i d to be an expansion of the  "Poincare t y p e " ([11], pp. 218-219) i f the sequence (x^ : n = 0, 1, 2, • • • ] , i s an asymptotic sequence (see D e f i n i -t i o n 2.1). When the range spaces B and B Q are i d e n t i c a l , $ c x » A convergent expansion i n terms of the elements of $ i s s a i d to be an expansion " e s s e n t i a l l y of the Poincare type". In [10] and [11] the d e s i r a b i l i t y , i n a theory of asymptotics, of a c c e p t i n g expansions which are not of the Poincare type i s h i g h l y s t r e s s e d . In an asymptotic space, 61 expansions which are not of the Poincare type can occur i f there e x i s t countable d i s t i n g u i s h e d sets with elements having a r b i -t r a r i l y s mall norms and which cannot be ordered i n t o an asymptotic sequence. S p e c i f i c examples w i l l be given i n the next s e c t i o n s . 3 - 4 Asymptotic spaces: Example I a) Our f i r s t examples of asymptotic spaces are simple and do not i n v o l v e any primary or secondary parameters: P = S = 0. The Hausdorff space A. i s the r e a l i n t e r v a l [0, A.], 0 < A, < <x>. and A. = 0. A cd-nbhd of 0 i s an i n t e r v a l of the o form (0, A,'], 0 < A,' < A,. The range spaces B and B Q are both the space of the r e a l numbers. N i s the set of a l l i n t e g e r s and the asymptotic s c a l e to be used i s $ = {«pn : n £ NJ, where q>n (A.) = A,n. Let X = fi be the space of a l l r e a l valued f u n c t i o n s x on A' which are a s y m p t o t i c a l l y f i n i t e with respect to $ and with norm d e f i n e d by ( 3 . 9 ) and ( 3 .10). C l e a r l y : $ c X; ? o(A.) s 1 and = 1; ^ - 0 ^ ) f o r a l l m, n £ N. From t h i s and Theorems 3°3 and 3 - 4 , i t f o l -lows that X i s a V - a l g e b r a . D i r e c t p r o o f s of t h i s r e s u l t were given by A. E r d e l y i [ 9 ] , J . Popken [ 2 9 ] / J- 0. van der Corput [ 3 8 ] , [ 3 9 ] . b) To i l l u s t r a t e the r e s u l t s of Chapters 1 and 2 , we c o n s i d e r the f o i l owing subsets of j^: / 62 (3-16) x" = l (Pa h • E = 1 a J = -a, < a < »}, n g N, n > k ] ; -« < a < »}, n = 0, 1, 2, ], wh (A.) - \ a ; ea(A.) = exp aX,; e re A. _ J (A.) i s the B.essel f u n c t i o n of the f i r s t k i n d n ( 3 . 1 7 ) ' z (A.) =• n 1 of order n; : n =0, 1, 2, •••}, where A,n s i n (n + l ) - ^ i f n i s even, A,n cos (n + l ) ^ i f n i s odd. It i s easy to v e r i f y that $ and <$^. are d i s t i n g u i s h e d subsets of ft. For n = 0, 1, 2, ( [ 8 ] , V o l . II) (3-18) Thus, by the Paley-Wiener Theorem (Theorem 2-2.U) J i s also a d i s t i n g u i s h e d subset of ft. The set G i s a l s o a d i s t i n g u i s h e d subset of ft since i t s elements have d i s t i n c t norms: |z I = o n . Since z = o(z ) i s • n • y n **• m not true f o r any n, m > 0, the sequence l z n ) i s not an asympto-t i c sequence. An expansion i n terms of the elements of G: ( 3 - 1 9 ) a s l n\ o A. f + a.A, cos-^f + a0A.2 s i n ^ f + 1 A. 2 A, a r e a l , n always converges to some element of ft. Yet, i t i s not an expansion of the Poincare type (see S e c t i o n 3# e ) . Expansions such as ( 3 . 1 9 ) are mentioned i n [ l l ] . 63 For a l l 8 * 0 , + 0z = 0(m ) i f and only i f m > rain[a, n ] . Ta ' z n ~ m Thus ( f , G) i s a d i s t i n g u i s h e d p a i r of subsets of R. A con-sequence i s that there e x i s t s a d i s t i n g u i s h e d b a s i s of ft which con t a i n s ^ and G. Another consequence i s that a f u n c t i o n which i s the sum of a s e r i e s ( 3 . 1 9 ) cannot admit an expansion i n terms of elements of ^, i , e . i n terms of powers of A,. The set E i s co n t a i n e d i n the c l o s e d subspace generated by § . Indeed, i t i s w e l l known that f o r any r e a l number a, ° r • a 1 1 the s e r i e s \ cpn converges a s y m p t o t i c a l l y , as A. -» 0, to the f u n c t i o n e^. E i s not a d i s t i n g u i s h e d subset of ft, since f o r a * B: I eal = X ' I e s l =  X' l e a " e s ' = P _ 1 < 1 -c) Let denote the c l o s e d subspace generated by G, i . e . the set of f u n c t i o n s which admit expansions of the form ( 3 . 1 9 ) . (See Theorem 1 -7-6) Let (P denote the c l o s e d subspace generated by i . e . the set of f u n c t i o n s which admit expansions of the form (3.20) a «> + a ffl + ... , n c N, a r e a l . w nyn n+lrn+1 ^ ' n (Pis a subalgebra of ft. A n o n - t r i v i a l element of 0* i s pseudo-r e g u l a r . From the above d i s c u s s i o n , ^- f\ (? is t r i v i a l . Let (P^ be the c l o s e d subspace generated by <$^ , i . e . the set of sums of expansions ( 3 .20) with n > k. E i s contained i n (P^ f o r a l l k < 0. 6 4 (Po i s a V - a l g e b r a . From (3. 18) and Theorem 2 - 2 . 4 , J i s a d i s t i n g u i s h e d b a s i s of (PQ. E and $ Q do not form a d i s t i n g u i s h e d p a i r of subsets of (Pq but t h e i r union i s a l i n e a r l y independent s e t . Therefore, there e x i s t s a Hamel b a s i s of (PQ which contains E \j $ o < In [ 4 ] , con-tinuous l i n e a r f u n c t i o n a l s on the subspace (E \J $ q) are s t u d i e d ; see Chapter 6. d) To c o n s t r u c t the space ft, we s e l e c t e d A , = 0 . C l e a r l y , we would have obtained a s i m i l a r space by choosing any other f i n i t e value f o r A , and the asymptotic s c a l e $ = lcpn : n g N ] where tp n(A0 = (X - A . Q ) N . One may a l s o c o n s i d e r ^ = [ A , , » ] , 0 < A . < «>, A . Q = 00 and the asymptotic s c a l e $ = : n £ N J , where (pn ( A , ) = A , N . e) Spaces such as ft can be c o n s t r u c t e d i n which X i s some sub-set of the complex plane and f o r more s o p h i s t i c a t e d asymptotic s c a l e s . Examples are given i n [11], with asymptotic s c a l e s such as tjl = { j n : n = 0, 1, 2, •••}, J n ( k , z,r,s) = T(z + ^) \~ -z-ns A . i s the complex asymptotic v a r i a b l e , A , A , = <»; z i s a complex number c o n s i d e r e d as a primary parameter, i . e . order r e l a t i o n s must h o l d u n i f o r m l y i n z; the p o s i t i v e r e a l numbers r and s are secondary parameters. Proper c o n d i t i o n s must be p l a c e d on the domains of A , and z. (See [11]; a l s o [ 9 ] , [10].) 65 3-5 Asymptotic spaces: Example II In t h i s s e c t i o n we g i v e two examples i n v o l v i n g formal power s e r i e s i n two r e a l or complex v a r i a b l e s . The spaces to be con-s t r u c t e d w i l l be used i n Chapter 4 to obtain asymptotic expan-sions of some f u n c t i o n s d e f i n e d as two-dimensional Laplace transforms [ 6 ] . b) In t h i s example the Hausdorff space A I s the set of p o i n t s A, = (u, v) of R f o r which 0 < u, v < m; \ q = (0, 0). The range spaces B and B q are the spaces of complex numbers and of r e a l numbers r e s p e c t i v e l y . N i s the set of a l l non-negative i n t e g e r s and the asymptotic s c a l e to be used i s § = {{pn : n c N}, q>n(u, v) = (u + v ) n . Let X = Q be the space of a l l complex valued f u n c t i o n s on A.' which are a s y m p t o t i c a l l y f i n i t e with respect to As f o r the space ft of S e c t i o n 4, one v e r i f i e s that Q i s a V-algebra, under the ^-asymptotic norm d e f i n e d by (3-9) and (3-10). Let x^ g X be the f u n c t i o n d e f i n e d by (u, v) = u*v j , i , j £ N . For a l l non-zero complex numbers ct, [3, and a l l i n t e g e r s I, j such that 0 < l < n, -l < j < n : (3.21) * . = 0(m ) i f and only i f m < n, T\-\ , i TH — (3.22) ax . . + f x . , = 0(m ) i f and only i f m < n. n ~ l t I rm ~ 66 Indeed, to ve r i f y ^  (3 • 2 2), suppose f i r s t l y that m < n. Then: | a u n - M .+ P u n ' J v J r < Max{|a|, | a | } . ( u n " V + u n " J v J ) < Max{|a|,|B|](u + v ) n . Thus : ax + l3x , = 0 L ) = O U ) n-l.I rn rm and ( 3 - 2 2 ) i s s a t i s f i e d i f m < n. Conversely, suppose that the r e l a t i o n i s true f o r some m = n + k, k > 0. Since every cd-nbhd of (0, 0) must contain p o i n t s (u, v) f o r which v = u, there e x i s t s a constant A > 0 such that f o r a l l u small enough i™ -l ft I n - n nn+k n+k | a + p | u < A • 2 u This i s i m p o s s i b l e . T h i s completes the v e r i f i c a t i o n of ( 3 - 2 2 ) . Consider the set H k = : 1 + j - k € N -It f o l l o w s from ( 3 . 2 1 ) and ( 3 . 2 2 ) that Hfc i s a d i s t i n g u i s h e d subset of Q . Let denote the c l o s e d subspace of Q generated by H^, i . e . the set of a l l f u n c t i o n s which admit expansions of the form ( 3 . 2 3 ) a .x , l• + j > k, a complex. I 3 *J i J < J U n l i k e the space (P^ of Example I (Section 4-c), the elements of a d i s t i n g u i s h e d b a s i s of do not have d i s t i n c t norms: from ( 3 - 2 l ) l xn-j,j I = f o r 0 < j < n. 67 A p a r t i c u l a r subspace of ^ s the subspace of f u n c t i o n s which admit expansions ( 3 ' 2 3 ) such that f o r some sequence i a n l of complex numbers ( i = - l ) : a . . = ( n ) i j a f o r 0 < j < n and a l l n. n-J,J j n - -S e t t i n g z = u + i v , the expansions of such f u n c t i o n s are of the V a z n, f orm n>k c) We now l e t the Hausdorff space A be a set of p o i n t s A, = (z, w) where z and w each belong to a subset of the complex Riemann sphere which c o n t a i n s the point at i n f i n i t y . Let \ = (oo, oo). ) N i s the set of a l l non-negative i n t e g e r s and we denote by ^ the asymptotic s c a l e [ t n : n € N}, t n ( z , w).« ( j ^ + - f a ) * . Thus, B o i s the space of the r e a l numbers. /Let B be the space of the complex numbers and c o n s i d e r the space Q' of complex valued f u n c t i o n s on A' which are asymptoti-c a l l y f i n i t e with respect t o ^ . It can be shown (as i n b)) that the set i s a d i s t i n g u i s h e d subset of Q* . X^. w i l l denote the c l o s e d subspace generated by H^. 3-6 Asymptotic spaces: Example I I I a) 63(D) w i l l denote the set of a l l bounded t r a n s f o r m a t i o n s 68 from a c l o s e d subset D of a Banach space S i n t o S i t s e l f ; i . e . the set of t r a n s f o r m a t i o n s from D to S f o r which fjAfJ^ < oo, where ||As -As || The f u n c t i o n ( 3 - 2 4 ) i s a pseudo-norm on dB(D). If ||A - B|| = 0, then As = Bs + S q f o r a l l s £ D and some f i x e d s e S. o Under the assumption that D i s a l i n e a r subspace of S, (D) w i l l denote the set of a l l bounded l i n e a r t r a n s f o r m a t i o n s from D to S. On D'CD.), ( 3 . 2 4 ) i s e q u i v a l e n t to s*0 and i s a norm under which 7^ (D) i s a Banach space ( [ 7 ] , p. 61; [26], p. 7 5 ) . The convergence induced by ( 3 . 2 4 ) on(B(D) and CT(D) w i l l be c a l l e d the uniform convergence on D ( [ 7 ] , p. 4 7 5 ; [26], p. 4 4 4 ) -Thus, a sequence [A } i n <B(D) converges u n i f o r m l y to A on D i f and only i f A 5 © ( D ) and l i m ||A-A || = 0, i . e . : n-4ao | | ( A - A n ) S l - ( A - A n ) s 2 | | l i m f i — ± rn = 0 f o r a l l s , , s 0 £ D, s, r" s 0, n^oo I I S l s2» 1 d 1 * b) We now c o n s t r u c t an asymptotic space of f u n c t i o n s with range i n <B(D), i . e . B = © ( D ) . 69 The Hausdorff space A i s the r e a l i n t e r v a l [0, l ] and A. = 0". The space B i s the space of the r e a l numbers. N i s o o the set of a l l non-negative i n t e g e r s and we use the asymptotic s c a l e $ = l<pn : n £ N}, cp n(^) = A--Let X be the space of a l l mappings x d e f i n e d on A T = (0, 1 ] , with range i n 0(D) and which are a s y m p t o t i c a l l y f i n i t e with r e s -pect to <$, i . e . such that f o r some n £ N, ( 3 . 2 5 ) ||x||D = 0 ( T n ) . Suppose that y £ X, y i s independent of A, and Hyflp ^ 0« Then, one v e r i f i e s e a s i l y that | y | = 1. Furthermore, i f x £ X s a t i s f i e s |y - x| < 1, then, as a f u n c t i o n of A,, x converges u n i f o r m l y to y on D, as A. -» 0, si n c e f o r A. small enough and some constant a > 0 |jy " x I I D < <*A.. If x ( X has a ^-asymptotic norm s t r i c t l y l e s s than 1 and f o r A. i n some cd-nbhd of 0, x maps D i n t o i t s e l f , then x(A.) i s a c o n t r a c t i o n mapping on D f o r a l l A, i n some cd-nbhd of 0, i . e . f o r some A,", 0 < A,' < 1, x(A.) maps D i n t o i t s e l f and j|x(A,)j|p < 1 when A, < A,' ( [ 1 9 ] , V o l . . I, p. 43) • Spaces of t h i s type have been considered by C. A. Swanson and M. S c h u l z e r [ 3 2 ] , [ 3 3 ] . I n these r e f e r e n c e s , t r a n s f o r m a t i o n s s a t i s f y i n g ( 3 . 2 5 ) are s a i d to be "of C l a s s L ip (cj>n)." S p e c i f i c examples are given i n [ 3 2 ] , pp. 2 8 - 3 8 . 70 c c) The space X of b) c o n s i s t s of mappings from ( 0 , l ] to the set (B(D) of bounded t r a n s f o r m a t i o n s from D to S. In the space X* t o be c o n s t r u c t e d now, unbounded t r a n s f o r m a t i o n s w i l l a l s o be co n s i d e r e d . The range space B i s the Banach space S. Let X, X Q , N and § be as i n b ) . Let X' be the space of a l l mappings from ( 0 , 1] x D to S which are a s y m p t o t i c a l l y f i n i t e with respect to $ when s £ D i s considered as a secondary parameter. The ^-asymptotic norm of x £ X' i s l e s s than or equal to p n i f f o r each f i x e d s £ D, there e x i s t a constant a [ s ] > 0 and a cd-nbhd V[s] of 0 such that \\x.(X, s)J| < a[ s]A. n f o r a l l A, £ V [ s ] ; e q u i v a l e n t l y , j x j < ^ n i f f o r a l l s £ D, ||x(A., s)|| < P[s]-A.n-I|s||, f o r a l l X € V [ s ] , where 3 [ s ] = a[ s ] • || sf| _ 1 i f s M . Suppose that y £ X f , y i s independent of A, and I"VI ^ 0 . Then f o r each s £ D, ||y(A., s) || = (|y(s)|| f o r a l l X £ and, hence, § y | = 1. Furthermore, i f x £ X» s a t i s f i e s |x - y | < 1, then, as a f u n c t i o n of X, x converges s t r o n g l y ( [ 7 ] , P- 475) to y on D, when X -» 0 ; indeed, f o r each s £ D, there e x i s t s 0.[s] > 0 such that f o r a l l X small enough ||x(A,, s) - y(A., s)|j < a[ s]-A.. 3-7 Moment spaces a) For s i m p l i c i t y we r e s t r i c t our d e f i n i t i o n of moment spaces to spaces of r e a l valued f u n c t i o n s d e f i n e d on a f i n i t e i n t e r v a l 71 [ a , b ] , -os < a < b < oo. A l l i n t e g r a l s c o n s i d e r e d a r e R i e m a n n -S t i e l t j e s i n t e g r a l s ( [ 4 1 ] , p . l ) • L e t a be a r e a l v a l u e d f u n c t i o n of b o u n d e d v a r i a t i o n on [ a , b ] ( [ 4 1 ] , P . 6 ) . L e t $ = l c p n ( t ) : n = 0 , 1, 2 , •••} be a s e q u e n c e of n o n - z e r o , r e a l f u n c t i o n s on [ a , b ] s u c h t h a t a l l i n t e g r a l s ( 3 . 2 6 ) n n ( l ) = J c p n ( t ) d a ( t ) , n = 0 , 1 , 2 , e x i s t and a r e f i n i t e . L e t X ' be t h e l i n e a r s p a c e o f a l l r e a l f u n c t i o n s x , d e f i n e d on [ a , b] a n d s u c h t h a t a l l i n t e g r a l s H n ( x ) = J x ( t ) c p n ( t ) d a ( t ) , n = 0 , 1 , 2 , a . e x i s t a n d a r e f i n i t e . U ( x ) i s c a l l e d t h e n - t h moment o f x r e l a t i v e t o (j). F o r x £ X ' , d e f i n e = i n f (n : ^ n ( x ) * 0} a n d , f o r some f i x e d p , 1 < p < « , ( 3 - 2 7 ) | x | = p - » ( x ) . I t i s i m m e d i a t e t h a t X f , w i t h t h e n o r m ( 3 » 2 7 ) r i s a V - s p a c e , e x c e p t p o s s i b l y f o r c o m p l e t e n e s s . I n X f t h e d i s t a n c e o f t w o f u n c t i o n s x , y i s l e s s t h a n o r e q u a l t o p n i f a n d o n l y i f H , , ( x ) = t t i ( y ) f o r i = 0 , 1 , 2 , n - 1 . X ' a d m i t s a d i s t i n g u i s h e d b a s i s ( T h e o r e m 2 - 2 . 2 ) . Two 72 elements of a d i s t i n g u i s h e d b a s i s of X* cannot have the same norm. Indeed, i f | x | = | y j = p n f o r some n, then ^ i ( x ) = ^.(y) = 0 f o r i < n; P<n(x) r" 0; P<n(y) £ 0 ; t h e r e f o r e H i ( ^ n ( y ) x - ^ n ( x ) y ) = 0 f o r i < n. T h i s i m p l i e s that | ^ n ( y ) x - ^ n ( x ) y | < p n . Thus x and y are not d i st i n g u i shed. Let N be the set of i n t e g e r s d e f i n e d by N = In : f o r some x c X f, u,(x ) = 6. f o r i < n}. 1 n ^ , r i n i n — For each n c N, l e t x be a f u n c t i o n such that LL. (x ) = 6. f o r " ' n I n i n i < n. The set H = [x^ : n £ N} forms a d i s t i n g u i s h e d b a s i s of X» . The completion X of X T, i . e . the set of formal expansions i n terms of H, i s a .V-space (Theorem 1-7.6). V-spaces c o n s t r u c t e d i n the manner d e s c r i b e d above w i l l be c a l l e d "moment spaces'*. b) For the remainder of the S e c t i o n we suppose that the func-t i o n a i s s t r i c t l y i n c r e a s i n g on [a, b] and that | i s a l i n e a r l y independent set of continuous f u n c t i o n s contained i n X. These assumptions imply that a l l the i n t e g r a l s (3.26) and the i n t e g r a l s .b J < p m ( t ) ' c p n ( t ) * d a ( t ) , m, n = 0, 1, 2, 9. e x i s t and are f i n i t e ( [ 4 1 ] , P« 7 ) . 73 A sequence [ f ^ l of f u n c t i o n s d e f i n e d on [a, b] i s s a i d to be orthonormal with respect to a i f <f , f > = 6 . m,n = 0, 1, 2, m n mn where .b <f, g> = [ f ( t ) . g ( t ) d a ( t ) . a Lemma 7«1» If f. i s a non-negative continuous f u n c t i o n on [a, b] and f ( t ) d a ( t ) = 0, then f (t) s 0 . J a The proof i s i d e n t i c a l to that of P r o p o s i t i o n ( 5 « 2 ) i n [ 3 7 ] , p . 4 1 » The orem 7 »2. There e x i s t s a unique sequence of f u n c t i o n s { p n l of the form p (t) = a en (t) + a i a > ..(t) + • • - + a m ( t ) , a > 0, n v nn Tn n , n - l T n - l noro ' nn ' which i s orthonormal with respect to a„ The proof i s an easy m o d i f i c a t i o n of [ 3 7 ] , pp. 4 1 - 4 2 . Theorem, 7 -3 • ( i ) |P n l = p'" > n = 0, 1, 2, ••• ( i i ) ( p j i s a d i s t i n g u i s h e d b a s i s of X. ( i i i ) If f ? X, then, i n the norm of X, (3-28) f ( t ) = Y < f ' P n > P n ( t ) n=0 Proof : ( i ) For a l l m, there e x i s t c o e f f i c i e n t s b , such that mi:. m < o ( t ) = V b . p . ( t ) , b = — ~ > 0 . rm v / mi I '' mm a . n mm l =0 74 Thus, f P n | = p f o l l o w s from (3.29) J P n ( t ) c p m ( r ) d a ( t ) • f 0 i f m < n, 1 i f m = n a . mm ( i i ) f o l l o w s from ( i ) and a previous remark, page 72, ( i i i ) For m < n, n-1 ^m( f - I < f ' P i > P i ) = i=0 n-1 = .J f ( t ) c p m ( t ) d a ( t ) - £ <f,p.> J P i ( t ) c p m ( t ) d a ( t ) a i=0 a m m = Y b .,<f,p,> - Y <f,P>b . = 0 . £ m i " i 1^ 1 mi Thus i=0 i=0 | f - ^ <f, P^P^ \ S p n w h i c h proves the convergence . The s e r i e s (3-28) i s u s u a l l y c a l l e d the F o u r i e r s e r i e s of f with respe ct to | p J ( [ 3 7 ] , P. 4 5 ) . In the moment space X, the d i s t a n c e of two f u n c t i o n s f and g i s l e s s than or equal to p n i f and only i f the f i r s t n F o u r i e r coef f i c i ents of f : <f,p^>, i = 0, 1, 2, n-1, are equal to the corresponding F o u r i e r c o e f f i c i e n t s <g, p^> of g. c) Whenever a i s s t r i c t l y i n c r e a s i n g on [a, b] and f o r a l l n, Cj>n(t) = [ c p ( t ) ] n , where (p i s a non-constant continuous f u n c t i o n on [a, b ] , the r e s u l t s of b) are v a l i d . Furthermore, we have the f o l l o w i n g Theorems 7.4 and 7.5. 75 Theorem 7-4' The orthonormal sequence {p ] s a t i s f i e s a recur-• ^ n rence formula of the form P n + l ( t } - t cn?-. ( t ) + d n K ^ + n * *V where c , d , e are r e a l c o n s t a n t s . (Set p n ( t ) s 0) . n n n -1 The proof i s a m o d i f i c a t i o n of the proof of P r o p o s i t i o n (5.4), [37], P. 43-The orem 7.5. B p m ° P n ^ — P a n c * s o m e c o e f f i c 1 e n " t s c m+n p ( t ) . p (t) = V c ,p. ( t ) . m n mm l mm * l=|m-n Proof ; Suppose m > n. Then b M ^ V ^ = [ P m ( t ) P n ( t ) T i ( t ) d a ( t ) a n +i = I" a m , r - i I P m ( t ) ? r ( t ) d a ( t ) , a r = l F rom (3»29), ^i^Pm'Pn^ = ^ f o r n + ^ < m « The c o n c l u s i o n f o l l o w s from t h i s and the f a c t that p mP„ i s a polynomial i n <p of degree m n m + n . For a d d i t i o n a l p r o p e r t i e s of the orthonormal sequence, F o u r i e r s e r i e s and c o e f f i c i e n t s , see [37], Chapter 5» d) Examples. Let ^>n("0 = t n and [a, b] = [-1, 1] . The f o l l o w -i n g -are three examples of moment spaces (See [37], P» 50. A l s o [ 8 ] , [21].) 76 da(t) ( 1 - t 2 ) 2 d t ( l + t 2 ) 2 d t dt P n ( t ) — T (t) «— n V TT n (t ) Reference T (t) : Chebyshev polynomials U (t) : Chebyshev polynomials of the second k i n d P ( t ) ; Legendre polynomials n To i l l u s t r a t e how a problem can be i n t e r p r e t e d w i t h i n the scope of a moment space, we c o n s i d e r the d i f f e r e n t i a l equation ( 3 - 3 0 ) Lx = 2(t + 3) jfi + x = 0, x(-l) = 1. Two methods have been proposed for. the approximation of the s o l u t i o n of a d i f f e r e n t i a l equation which take advantage of the s p e c i a l p r o p e r t i e s (given i n b) and c) above) of the Chebyshev pol y n o m i a l s . One i s due to Lanczos [ 2 0 ] , [ 2 1 ] , the other to Clenshaw [ 2 ] , See L. Fox [ 1 3 ] . In both methods, the equation ( 3 . 3 O ) i s r e p l a c e d by the equation ( 3 . 3 1 ) My(t) = Ly(t) - TT r (t ) =0, y(-l) = 1, It can be shown that f o r a c e r t a i n value T of the para-o me t e r T, ( 3 « 3 l ) has a s o l u t i o n of the form ( t ) - T o I S . T . t t ) . w i t h T „ - ( J ( - D ^ J - 1 i = 0 By Theorem 7.5, i f z £ X, then Lz and Mz belong to X and Lz - Mz I < p -n 77 A.study of t h i s method of s u b s t i t u t i o n of a perturbed equation f o r the o r i g i n a l one, i f conducted w i t h i n the frame of the theory of moment spaces, may lead to i n t e r e s t i n g r e s u l t s and i n t e r p r e t a t i o n s . 78 CHAPTER 4 BOUNDED OPERATORS ON V-SPACES 4-1 D e f i n i t i o n s and n o t a t i o n s In t h i s Chapter, unless otherwise s p e c i f i e d , X and Y w i l l denote two V-spaces over the same f i e l d of s c a l a r s F; Z w i l l denote a c l o s e d subset of X. An operator from Z to Y i s a s i n g l e valued mapping d e f i n e d on a l l of Z with range i n Y. The conventions of pages 31 and 45 apply i n t h i s Chapter. D e f i n i t i o n 1 . 1 . An operator A from Z to Y i s s a i d to be 1inear  on Z i f A(ciu + Bv) = aAu + BAv f o r a l l a, 3 £ F and a l l u, v ^ Z such that au + 3v £ Z . D e f i n i t i o n 1 . 2 . Let A be an operator from Z to Y. ( i ) The nornJ of A on Z , denoted by I A J ^ i s d e f i n e d by ( 4 . 1 ) |A| Z = i n f { M > 0 : | A u - A v l < M|U-V| f o r a l l u,v £ Z J ( i i ) If Z = X, the norm of A on X i s denoted by |AJ, i . e . |A| X = JA|. ( i i i ) A i s s a i d to be bounded on Z i f | A j ^ < ». It f o l l o w s that j A |^ = 0 i f and only i f f o r some f i x e d y £ Y and a l l u £ Z , |Au - y | = 0 . If Z i s a l i n e a r subspace of X and A i s l i n e a r on Z , ( 4 - l ) i s e q u i v a l e n t to 7 9 ( 4 . 2 ) f A | z ; = i n f { M > 0 : |Au| < Mju| f o r a l l u £ z}. In X, the b a l l s S(9, r ) , S'(9, r) are c l o s e d subspaces of X and the q u o t i e n t spaces X/S(9, r ) , X/S f(9, r) are d i s c r e t e V-spaces (See Theorem 1-2.1). Consequently, the norm on X of an operator, even a l i n e a r operator, cannot be determined by con-s i d e r a t i o n of i t s values on these b a l l s only (unless, of course, X = S(0, r) or X = S'(9, r ) ) . This i s in s t r i k i n g c o n t r a s t to the case of a l i n e a r operator on a Banach space ( [ 7 ] , [ 3 ° ] ) where fjA|| = inf{M > 0 : |JAxJ| < Mf|x||for a l l x £ S»(9, r)} In a V-space, i f Z D Z», then | A | 2 > | A J z , . (Z, Y) w i l l denote the set of a l l bounded operators from Z to Y. *3 (Z, Y) w i l l denote the set of a l l bounded l i n e a r opera-t o r s from Z to Y. If Z = X = Y, we s h a l l use the n o t a t i o n s ©'(X) and CT(X) in plac e of CT(X, X) and J ( X , X). The product AB of two elements A, B of C^(X) i s d e f i n e d by (AB)x = A(Bx) f o r a l l x £ X. It i s simple to v e r i f y that ( 4 - 3 ) | A B | < | A | - I B | . . In g e n e r a l , the product of n o n - l i n e a r operators does not s a t i s f y c o n d i t i o n s (2.9) and (2.10) of D e f i n i t i o n 2-6.1, and hence 0~(X) i s not an a l g e b r a . These c o n d i t i o n s are s a t i s f i e d f o r l i n e a r operators and hence tJ (X) i s a non-commutative sub-a l g e b r a of the space CT(X). (See [7], [36].) 80 0 w i l l denote the zero-operator i n CT(Z, Y) : Ou = 9 £ Y f o r a l l u £ Z. I w i l l denote the i d e n t i t y operator i n i . e . Ix E x f o r a l l x £ X. 4-2 The spaces Q*(Z, Y) and S*"(X) of bounded operators The spaces GT(Z, Y) and 0~(X) are l i n e a r spaces over the f i e l d of s c a l a r s F. C l e a r l y the elements of CT(Z, Y) or of GT(X) 'are continuous mappings on t h e i r domains of d e f i n i t i o n , Z or X . The norm on Z, d e f i n e d by (4-1) has the f o l l o w i n g proper-t i e s : ( i ) In accordance with convention ( i i ) , page 3 1 , both the norms on X and on Y are expressed i n terms of the same r e a l num-ber p ( D e f i n i t i o n 2 - 1 . 1 ) . It f o l l o w s that the norm of an operator i n 0~(Z, Y) has a norm equal to zero or to ^  n f o r some i n t e g e r n. ( i i ) |aA| z = IA|z f o r a l l A f GT(Z, Y) and a l l a £ F, a * 0. ( i i i ) | A + B| z < Max{|A| z, IB |z} f o r a l l A, B € CT(Z, Y). Indeed, f o r a l l u, v £ Z: §Au + Bu - Av - Bv| < Max[|Au - Av|, 1'Bu - BvJ} < (Max[|A| z, |B| z})|u. - V|. ( i v ) [A + B| z = Max[|A| z, | B| z] whenever | A | z * \ B | z . To prove t h i s , suppose without l o s s of g e n e r a l i t y that |A| > |B | . Then, f o r every e > 0 such that 81 0 < £ < (|A|Z - | B | Z ) , there e x i s t u = u(e) and v = v ( e ) , i n Z, such that |Au - Av|~> (|A|Z - e)|u - v | > |B| z.|u - v | > |Bu - Bv |. Thus, |Au + Bu - Av - Bvf = |Au - Av|. It f o l l o w s that f o r every £ > 0, |A + B| z > | A J Z - £, G A + BS Z = | A | Z = M a x ( L A | Z , | B | Z J . These r e s u l t s lead to the f o l l o w i n g theorem on the s t r u c -ture of QT(Z, Y), (and of 0*(X) when Z = X = Y). \ Theorem 2.1. The space C ( Z , Y), under the norm on Z d e f i n e d by (4.1), i s a V-space. Proof : It f o l l o w s from ( i ) - ( i v ) above that CT(Z, Y) s a t i s -f i e s a l l the d e f i n i n g p r o p e r t i e s of V-spaces, except p o s s i b l y f o r complet ene s s. To prove the completeness of 0~(Z, Y), consi d e r a Cauchy sequence (A } i n 0"(Z, Y ) . S i nee (Th. 1-4 «1) lim |A n + 1 - A j z - 0, n_,oo f o r any £ > 0, there e x i s t s an i n t e g e r N ( E ) such that f o r a l l u, v ^ Z and a l l n > N ( e ) , I (A A . u - A - (A u - A v ) i < £ju - v I. • n + 1 n+1 n n • • 1 82 Let us s e l e c t an a r b i t r a r y p o i n t X q of Z. F o r each x £ Z, th e s e q u e n c e [A x - A x } i s a Cauchy sequence i n Y; s i n c e Y i s _ n n o c o m p l e t e , t h i s s e quence has a l i m i t . Let A be an o p e r a t o r from Z t o Y d e f i n e d by Ax = 1 i m ( A x - A x ) , x f Z . n n o *^ n_»oo We s h a l l show t h a t A i s a l i m i t of iAn1• F o r u, v g Z, d e f i n e y (u, v) == [ (A u — A x ) — (A U - A X ) ] Jn,p L n + p n + p o n n o J - [ ( A v - A x ) - ( A v-A x )] . L V n + p n+p o' v n n o / J F o r any £ > 0, t h e r e e x i s t s N ( & ) such t h a t f o r a l l u, v £ Z, a l l n > N(e) and a l l p > 0, | y n , p U , v ) | = | ( A n + p u - A n u ) - ( A n + p v - A n v ) | * K+p " Anlz'lU " V l < £»U " 'I-With n f i x e d , we have, f o r a l l u, v g Z: l i m y (u, v) s (Au - A u) - (AT - A V ) n, p / = \ n / n P-4CO S i n c e l i m l y (u, v ) | = I l i m y (u, v ) | , we -have, f o r • n,p • • n,p ' P_» 00 P-i 00 n > N ( E ) and a l l u, v ^ Z: | (Au - A n u ) - (Av - A v ) | < e|u - v|. Hence l i m | A - A n | z - 0 n-«oo T h i s shows t h a t A i s a l i m i t of t h e sequence i ^ } . As l i m i t of a Cauchy sequence A i s bounded on Z and 83 |A| Z " l i m | A n | z We note that the operator A d e f i n e d above depends on the s e l e c t e d p o i n t x 0 . C l e a r l y two d i f f e r e n t s e l e c t i o n s of X q w i l l i n g e n e r a l generate two d i s t i n c t l i m i t s f o r the sequence the norm of the d i f f e r e n c e between two such l i m i t s i s obviously zero. Remark. From ( i i ) , ( i i i ) , ( i v ) , pp. 8G, 81, and the f a c t that the proof of the completeness of 0"(Z, Y) r e q u i r e s the complete-ness of Y only, we can deduce t h a t : ( i ) If X and Y are (pseudo-) valued spaces, then ©*(Z, Y) i s a pseudo-valued space; ( i i ) If X i s a (pseudo-) valued space and Y i s a s t r o n g l y (pseudo-) valued space, then 0"(Z, Y) i s a s t r o n g l y pseudo-valued space. ( i i i ) If Y i s complete, then 0*(Z, y) i s complete. 4~3 The spaces tJ(Z, Y) and tf (X) of bounded l i n e a r operators ^f. (Z, Y) i s the set of bounded l i n e a r operators from Z to Y. To avoid meaningless or t r i v i a l statements we s h a l l assume that Z p r o p e r l y lends i t s e l f to l i n e a r i t y arguments. This i s achieved by r e q u i r i n g that Z be l i n e a r l y n o n - t r i v i a l , , d e f i n e d as f o l l o w s : A subset Z of a l i n e a r space i s ' s a i d to be l i n e a r l y n o n - t r i v i a l i f and only i f there e x i s t u, v £ Z and a, B £ F such that z = a u + 0 v £ Z , z ^ u , z $ v a n d | z | ^ 0 . 84 Obviously, any n o n - t r i v i a l subspace of a V-space i s l i n e a r l y non-t r i v i a l . Theorem J > . \ . The space ^ (Z, Y) i s a V-space. The space *3 (X) i s a V - a l g e b r a . Proof ; Using c o n t i n u i t y of the operators i n v o l v e d , i t i s easy to v e r i f y that *J (Z, Y) is. a c l o s e d l i n e a r subspace of &~ (Z, Y). In 7T (X), the product of two l i n e a r operators i s a l i n e a r opera-t o r . Then, the theorem i s a c o r o l l a r y of Theorem 2 .1 . The f o l l o w i n g theorems are analogous to theorems v a l i d i n t o p o l o g i c a l normed l i n e a r spaces over the r e a l or complex f i e l d s with t h e i r u s u a l v a l u a t i o n s . The proofs are s i m i l a r to those of the corresponding theorems i n [36], PP • 18, 85-86, and are omitted. We s h a l l use the f o l l o w i n g d e f i n i t i o n : D e f i n i t i o n 3 .2 . Let A g C f ( Z , Y) . An operator A - 1 from A(z) to Z i s c a l l e d a pseudo-inverse of A on A(z) i f A ^(Az) = z f o r a l l z £ Z. Theorem 3-3- If Z i s a subspace of X, then A £ ^ (Z, Y) i s con-tinuous e i t h e r at every point of Z or at no poin t of Z. The orem 3 • 4 • Let Z be a subspace of X and A £ * J (Z, Y). ( i ) A pseudo-inverse of A on A(Z), when i t e x i s t s , i s l i n e a r on A ( z ) . ( i i ) A admits a bounded pseudo-inverse on A(z) i f and only i f there e x i s t s a constant m > 0 such that m|z| < | A z | f o r a l l 8 5 z e z . A l i n e a r operator from a Banach space to another i s bounded i f and only i f i t i s continuous ( [ 3 6 ] , p. 8 5 ) . In V-spaces boundedness i m p l i e s c o n t i n u i t y but the converse i s not t r u e . (See Example 1 , p. 8 6 ) . A. F. Monna ( [ 2 / + ] , P a r t I I I , p. I I 3 6 ) has proved that l i n e a r operators from a V-space to i t s f i e l d of s c a l a r s F, considered as a V-space over i t s e l f , are continuous i f and only i f they are bounded. The f o l l o w i n g theorem g e n e r a l i z e s t h i s r e s u l t ; the proof i s modelled a f t e r that of Monna. Theorem 3 - 5 . Let A £ ^ ( Z , Y) and suppose that A(Z) i s a d i s -c r e t e and bounded subspace of Y. Then, A i s bounded i f and only i f i t i s continuous. Proof ; Boundedness i m p l i e s c o n t i n u i t y . To prove the converse, suppose that A i s continuous. Since A(Z) i s a d i s c r e t e subspace, there e x i s t s £ > 0 such that (4.4) y 6 A ( z ) an<* 1Y J < e imply | y | = 0. Since A i s continuous, there e x i s t s 6(e) such that z £ Z and | z j < 6(e) imply |Azg < e, and t h e r e f o r e , by (4»4) z £ Z and | z | < 6(e) imply |AzJ = 0. Since A(z) i s bounded, there e x i s t s M > 0 such that | y I < M f o r a l l y ^ A ( Z ) . 8 6 For a l l z £ Z such that | z | > 6 ( e ) , we have | A z | < M = y ( f j • 6 ( e ) < ^ | z | • • M Hence, l ^ f ^ < ^ ( £) , and A i s bounded. We conclude t h i s s e c t i o n with two examples. The f i r s t i s an example of a continuous unbounded l i n e a r operator from a V-space to i t s e l f ; the second shows that the Uniform Boundedness P r i n c i p l e ([36], p. 2 0 4 ; [7], P» 6 6 ) does not h o l d i n V-spaces, i . e . a family of bounded l i n e a r operators on a V-space which i s p o i n t -wise bounded i s not n e c e s s a r i l y u n i f o r m l y bounded. Example 1. Let X be a V-space over the r e a l numbers, with a countable d i s t i n g u i s h e d b a s i s H = {h , h^, hg, •••} such t h a t , f o r some i n t e g e r k, | h n f = p"" k _ n, n = 0, 1, 2, . (The space (P^ of 3 - 4 i s such a space.) Every n o n - t r i v i a l element of X has an expansion i n terms of H: (*'5> X = J ( a n h 2 n + hh2n+l)> V P n € R ' n =N where N > 0 and | a N | + | 3 N | * 0 . Let A be an operator from X to i t s e l f d e f i n e d by Ax = « 9 i f | x | = 0 , 00 Y (a + B )h i f x i s given by (4.5'). * V1-. n ri ri n=N 87 C l e a r l y A i s l i n e a r . A i s unbounded s i n c e | A h 2 n | = p n | h 2 n | f o r a l l n = 0, 1, 2, Yet, given any i n t e g e r n > 0, A ( S ( 0 , - k - 2 n p )) c S(9, p - k _ n ) . This shows that A i s continuous at 9 and, from Theorem 3-3, that A i s continuous on a l l of X. Example 2 . Let X be as i n Example 1. Every n o n - t r i v i a l element x of X has an expansion i n terms of H: w (4.6) x = £ a nh n, a N * 0, N > 0 . n=N For each non-negative i n t e g e r p, l e t A^ be an operator from X t o i t s e l f d e f i n e d by ( 9 i f | x | = 0, A x = P V a h L n n - i i f x i s given by (4.6), N > p, n=N P - 1 ^ a n n D + ^ a n h n - p i f X i s ^ i v e n b y (4.6), N < n=N n=p i.e.: the image of h i s h i f n < p and i s h i f n > p, * n o — n-p — The l i n e a r i t y of A i s e a s i l y v e r i f i e d . We have P | A p x | = P - K " N + P = p " | x | i f N > p i n ( 4 . 6 ) . -k P ' P P < p P | x | i f N < p i n (4-6) |A x l • P • Hence: | A P | = p P , p = 0, 1, 2, ••• 8 8 T h i s shows that the f a m i l y of l i n e a r operators U J i s a fa m i l y of bounded l i n e a r operators which i s not uniformly bounded , . P s i n c e lira A = o>. Yet, the f a m i l y (^  1 i s poin t - w i s e bounded since | A p x j < p ^ f o r a l l x £ X. We have shown that the Uniform Boundedness P r i n c i p l e does not h o l d i n V-spaces. ([36], P• 2 0 4 . ) 4 ~ 4 C h a r a c t e r i z a t i o n of bounded l i n e a r operators In t h i s s e c t i o n X and Y are V-spaces, Z i s a l i n e a r sub-space of X which i s not n e c e s s a r i l y c l o s e d , H i s a d i s t i n g u i s h e d b a s i s of Z. With each element h £ H, l e t there be a s s o c i a t e d an element Ah £ Y such that f o r some M > 0, I Ah I < Mjh I f o r a l l h f H, ( 4 - 7 ) §Ah I = Mlh I f o r some h c H. • o • * o• o . Each z £ Z i s a sum of an expansion i n terms of H: (4.8) z = a 1 h 1 + a 2 h 2 + ... , a. ^ F, a. f |h. | > | h i + 1 1 If [h^, h 2 , ...] i s i n f i n i t e , then l i m I N N § = 0 and, by n-»ao (4.7), l i m j A h n | = 0. n_.oo We extend the d e f i n i t i o n of A to a l l of Z, by s e t t i n g (4.9) Az = QC1Ah1 + Ct 2Ah 2 + «»• , z given by (4.8). 89 Since Y i s complete, t h i s s e r i e s converges and Az i s de f i n e d , up to a d d i t i o n of t r i v i a l elements. C l e a r l y , A i s a l i n e a r operator from Z to Y. It i s al s o bounded s i n c e , by Lemma 1 - 7 . 5 , | A z | < sup[JAh n|} < M-sup[|h n|} = M j h ^ = M|z|. n n In view of ( 4 - 7 ) , |A| = M. We have c o n s t r u c t e d an element of tT (Z, Y ). It i s impor-tant to note that the values of A on H were completely a r b i t r a r y , except f o r c o n d i t i o n s ( 4 » 7 ) . Now, suppose that B i s a continuous l i n e a r operator and that Bh - Ah £ [©] f o r a l l h £ H. Then Bx - Ax £ [9] f o r a l l x £ (H); ( r e c a l l that (H) i s the set of a l l f i n i t e l i n e a r combinations of element s of H). Since (H) i s dense i n Z and B - A i s continuous, we must h ave Bz - Az £ [9] f o r a l l z £ Z. This r e s u l t can be s t a t e d as f o l l o w s : Theorem 4 * 1 . Let X and Y be a V-space and l e t Z be a l i n e a r sub-space (not n e c e s s a r i l y c l o sed) of X. ( i ) An elemen t A of 7 (Z, Y) i s determin ed, up to a d d i t i o n of t r i v i a l elements, by i t s values on a d i s t i n g u i s h e d b a s i s H of Z, and 9 0 |A | z = inf{M > 0 : |Ah| < M|h| f o r a l l h £ H}. (ii) If a s i n g l e valued mapping A i s . a r b i t r a r i l y d e f i n e d on H except f o r the requirement that ( 4 . 1 0 ) s u p j ^ j i be f i n i t e , then A can be extended by l i n e a r i t y to a l l of Z and J A j ^ i s equal to (4.10). Furthermore, i f B i s a continuous l i n e a r operator from Z to Y and |Bh - Ah| = 0 f o r a l l h £ H, then B g J ( Z , Y) and [B - A| z = 0 . A p p l i c a t i o n 1 . The important f e a t u r e of the a s s e r t i o n of part ( i i ) of the above theorem i s t h a t , p r o v i d e d ( 4 » 1 0 ) i s f i n i t e , the values of A on the elements of H are a r b i t r a r y . The same i s not true i n an i n f i n i t e dimensional H i l b e r t space X i n which H = {h^, h,,, h^, •».] would represent a countable complete orthonormal b a s i s . As two examples, c o n s i d e r A and B d e f i n e d on H by ( 4 . 1 1 ) Ah = h. for all n :, n 1 ( 4 . 1 2 ) Bh = h . for 2 1 - 1 < n < 2 1 - 1. Then sup hgH Ahfl II Bh = I < However, n e i t h e r A nor B can be extended by l i n e a r i t y to a l l of X: they are not d e f i n e d at the poi n t S = ^ *, - h . £,n = l n n 91 If we suppose now that X i s a V-space with d i s t i n g u i s h e d ' b a s i s H = (h^, h^, •••.} and l i m = 0, then the mapping A of n_»m 1A (4.11) i s not acceptable under Theorem 4*1 s i n c e sup -"-r h^H 1 the mapping B can be extended i n t o an element of T (X) since Igh I , sup 1 j " I = 1. hfH 1 1 A p p l i c a t i o n 2. Theorem 4-1 f i n d s an a p p l i c a t i o n i n a paper of H. F. Davis [ 4 ] « We present the problem of [ 4 ] i n our own terminology. The n o t a t i o n i s that of S e c t i o n 4 , Chapter 3-Let X = Y = Let Z be the open subspace of ft f o r which the set $ o y E (see (3-16)) i s a Hamel b a s i s . Let A' be a s i n g l e valued mapping d e f i n e d on $ \J E by A'"?n 2 P( n)<p n» n = 0, 1, 2, A ' e a H f a , -a, < a < oo, a £ 0, where B(n) i s a s c a l a r and f i s an element of ft. S i n c e every element of Z i s a unique f i n i t e l i n e a r combina-t i o n of the elements of <$q [) E, A' can be uniquely extended by l i n e a r i t y to a l l of Z. Let B denote t h i s e x t e n s i o n . The main theorem of Davis [ 4 ] a s s e r t s that a necessary and s u f f i c i e n t c o n d i t i o n f o r B to be continuous on Z i s t h a t , i n the (j>-asymptotic norm on ft: 00 ( 4 - 1 3 ) f a = Y a n | a n ) ¥n' f o r a 1 1 a * °-n=0 The r e s u l t i s obtained from Theorem 4-1 through the 92 f o l l o w i n g argument : (p i s a d i s t i n g u i s h e d b a s i s of Z c see S e c t i o n 3 ~ 4 ) . The mapping A' i s d e f i n e d on 0 i n such a way that sup —a—Hr- = 1. ° n>0 ITn I Thus, by Theorem 4 . l ( i i ) , there e x i s t s a "unique" l i n e a r operator A from Z to ft which agrees with A* on $ q; t h i s extension A i s such that | A | = 1 and s i n c e ,n n=0 we must have 00 n A e a I n l A Tn 2- nl *n" n=0 n=0 Hence, the above l i n e a r operator B i s continuous on Z i f and only i f |Bz - A z j = 0 f o r a l l z £ Z; i . e . B i s continuous i f and only i f f = B e a = Ae a, so that ( 4 . 1 3 ) i s s a t i s f i e d . The reader w i l l n o t i c e that the d e f i n i t i o n of c o n t i n u i t y which we use and the d e f i n i t i o n of "asymptotic c o n t i n u i t y " given by Davis ( [ 4 ] , P» 91) are d i f f e r e n t . Keeping ,to our own terminology, Davis c a l l s an operator A " a s y m p t o t i c a l l y c o n t i n u -ous" i f | x | < p n i m p l i e s |Ax| < p n . T h i s i s e q u i v a l e n t to s a y i n g that A i s continuous i f and only i f | A |^ < 1. However, in the example above |A |^ = 1 and the two d e f i n i t i o n s of con-t i n u i t y l e a d to i d e n t i c a l r e s u l t s . T h i s d e f i n i t i o n of "asymptotic c o n t i n u i t y " i s too r e s t r i c -t i v e . An operator which i s a s y m p t o t i c a l l y continuous i s a l s o continuous i n the t o p o l o g i c a l sense but the converse i s not t r u e . 93 The d e s i r a b i l i t y of removing such r e s t r i c t i o n s i s comparable to the d e s i r a b i l i t y of a c c e p t i n g asymptotic expansions which are not of the Poincare type (see S e c t i o n 3 -3# e^» App l i c at i on 3 • Define the l i n e a r operator L from the space ^ o to the space X ^ of S e c t i o n 3~5 by: 00 00 ^ ( L x ) ( z , w) = j J e~ V x ( u , v)du dv. o o S i n c e and Lx . =.mI mn n I ym + l , n + l ' m ' n * °' -2. |Lx 1 = D | X I, f o r a l l m, n > 0, 1 mn ' P • mn ' — L can be extended (by Theorem 4 • 1) to a l l of X Q and | L | = p *~. I f , i n the asymptotic norm on X , the f u n c t i o n x admits o the expansion oo n x = } Y a J X • • i L L n,j n - j , j ' n=0 j-0 ' then, i n the asymptotic norm on <X ' , oo n L X " I I ° a . J . ^ - J ) , J | ' . - J * l . j * l • n=0 j=0 This r e s u l t p r o v i d e s means to obtain asymptotic expansions of Laplace transforms i n two v a r i a b l e s . See [ 6 ] . For example, s i n c e J (^Tv) has the asymptotic expansion: so n n J_(/Tv) = V v p , n t 0 2 2 n ( n l ) 2 9 4 03 LJ o(./uv) zw I 1 4zw-l (4zw) n n=0 (Compare with [ 6 ] , p. 100.) 4-5 Inverses and s p e c t r a i n (X) The V-space "^7 (X) i s a V - a l g e b r a . In accordance with D e f i n i t i o n 2 - 6 . 2 , a p s e u d o - i d e n t i t y i n *J (X) i s a l i n e a r operator I1 such that |l» - 11 = 0 . The d e f i n i t i o n of a pseudo-inverse A 1 of A on i t s range A(X) has been given ( D e f i n i t i o n 3 - 2 ) . The operator A 1 belongs to (X) i f and only i f i t i s bounded and d e f i n e d on a l l of X. T h e r e f o r e , A i s (pseudo-) r e g u l a r i n the sense of D e f i n i t i o n 2 - 6 . 3 i f and only i f i t admits a bounded (pseudo-) i n v e r s e and A(X) = X. In such a case, we s h a l l say that A admits a (pseudo-) i n v e r s e A \ without any mention of the range of A. Let A, s A - A.I. By D e f i n i t i o n 2 - 6 . 7 , \ belongs to the A. spectrum o-(A) of A i f and only i f A^ i s s i n g u l a r . Theorems 6 . 6 , 6 . 8 and 6 . 9 of Chapter 2 apply to the V-a l g e b r a 3 (X), ( with x £ X r e p l a c e d by A £ 3" (X) and e r e p l a c e d by I ) . The f o r m u l a t i o n of these three theorems f o r bounded l i n e a r operators on a V-space should be compared with s i m i l a r theorems f o r bounded l i n e a r operators on Banach spaces: see [36], Theorems 4 » 1 - C and 4-1-D, page 1 6 4 , and Theorem 5.1-A, page 2 5 6 . 95 Note: As i n Theorem 5•1-A of [ 3 6 ] , we can add to the statement of Theorem 2 - 6 . 9 the f o l l o w i n g p r e c i s i o n : Let A £ 3 (X)» I -A. I > 1; i f f o r some s c a l a r ^ g F, A has a pseudo-i n v e r s e A on i t s range A (X) and |A ''"I < 1, then, f o r a l l A. £ F, A^ has a pseudo-inverse on i t s range A^(X) and the topo-l o g i c a l c l o s u r e of the range of A^ i s not a proper subset of the t o p o l o g i c a l c l o s u r e of the range of A . The proof i s i d e n t i c a l to that given i n [ 3 6 ] , p. 2 5 6 . Our m o d i f i c a t i o n of Ri e s z ' s Lemma (Theorem 1^6.1) must be used. 4-6 Complete s p e c t r a l decompositions The s c a l a r A, i s c a l l e d an eigenvalue of A ^ J (X) i f f o r some x^ £ X, 4= 0, Ax^ = ^*xx,° T ^ e P°i n" t x ^ i s c a l l e d an eigenelement a s s o c i a t e d with A.. The set X^ = (x £ X : Ax = A.x} i s a c l o s e d subspace of X and i s - c a l l e d the eigenspace a s s o c i a t e d with A,. D e f i n i t i o n 6.1. An operator A £ H (X) i s s a i d to have the complete s p e c t r a l decomposition { (A,^, h^) : i ^ j ] i f f o r each i i n the index set J , Ah^ = ^ h ^ » not a l l A.^  are equal to 0 and the set of eigenelements H = {h^ : i f j} i s a d i s t i n g u i s h e d b a s i s of X. Theorem 6.2. If A has a complete s p e c t r a l decomposition I (A-i, h±) 1 i £ J}, then : 96 (i) |A| = 1; (ii) For a l l X. ^ (A,^  : i ( j } , A^ i s pseudo-regular and (iii) If A.^  ^ 0 f o r each i £ J , then A i s an isometry on X, i.e. j A x | = |x| f o r a l l x £ X. Proof; ( i ) The operator A s a t i s f i e s (4.7) with M = 1. Thus, by Theorem 4.1, |A| = 1 . (ii) Let x be an a r b i t r a r y p o i n t i n X. It admits a non-i n c r e a s i n g expansion i n terms of H: 80 ( 4 . 1 4 ) x = £ a n h n , a n € F, a Q * 0, h n € H. n=0 By Lemma 1 - 7 . 5 , | x | = | h. ^  |. Then, CO ( 4 . 1 5 ) A, x = V a (X - X)h . A, £ n n n n=0 If A. ^  [A^ : i £ j } , |A^x| = | h. Q | = |x|. It f o l l o w s from Theorem 4 » 1 # that the operator A ^ \ d e f i n e d on H by A r l h ' = T ^ T h. , h. c H, X I X.-X i ' i ^ ' 1 i s a pseudo-inverse of A^. If x i s given by (4.I4), so A ^ X = y a -r-i-r- h . A, / n A. -X n n ~0 Thus, |A^x| = | h j = |x|. ( i i i ) f o l l o w s from (4.I4) and ( 4 » 1 5 ) with A. = 0, A,n t 0 f o r each n £ J . 97 C o r o l l a r y 6 . 3 . If A € (X) admits a complete s p e c t r a l decomposition, then the c a r d i n a l i t y of the set of i t s e igen-values cannot exceed the dimension of the space. Proof ; Let { (A,^, h^) : i £ J} be a complete s p e c t r a l decomposition of A. If the c a r d i n a l i t y of the set of e i g e n -values exceeds the dimension of the space, i . e . the c a r d i n a l i t y of the d i s t i n g u i s h e d b a s i s {h^ : i £ J j , there e x i s t s an eigenvalue A, which does not belong to fA.^  : i £ j } . Since A. i s an eigenvalue, A., i s s i n g u l a r . This c o n t r a d i c t s ( i A. of Theorem 6 . 2 . In the f o l l o w i n g Lemma 6.4 and Theorem 6 . 5 / the assump-t i o n s and n o t a t i o n s are as f o l l o w s : A £ ZJ (X) admit s a complete s p e c t r a l decomposition [ (A > i, h.) : i £ j } . H = [h. : i £ j } . For an a r b i t r a r y s c a l a r A,, .= [ i £ J : A.. = A,}, = [h. £ H : A,. = A.}. X^ denotes the c l o s e d subspace generated by H^„ C l e a r l y , i f A, i s not an e i g e n v a l u e , by Theorem 6 . 2 ( i i ) , A. ^ A,^  f o r each i £ J and, t h e r e f o r e , J ^ , and X^ are empty. If A. i s an ei g e n v a l u e , then A, = A.^  f o r some i £ J and X^ i s the non-empty eigenspace a s s o c i a t e d with A.. Let P^ denote a l i n e a r operator from X to X, d e f i n e d on H by h i f h £ H^, 9 i f h £ H\H. A," 1 98 By Theorem 4-1, |P^| = 1 i f X^ i s not empty and |P-^| = 0 i f X^ i s empty. Lemma 6.4- For a l l x £ X and a l l s c a l a r s A.: I x -- P^x| = I Ax - A,x|0 Proof; Given x £ X, x admits an expansion of the form x = I -(*' V H V + I ( x< h i } H h i h i ^ H 7 , h i ^ \ (For n o t a t i o n , see p. 28)- Thus, |x - P^xf = 1 Y ( x ' h i > H hJ h . £HNH, 1 A. |Ax - A.x| = I Y (\ ~ X'(x> hi^E h i l 1 A. By Lemma 1-7•5, |x - P^x fj = j Ax - A,x| = 0 i f (x, h . )„ = 0. f o r a l l 1 n h , £ H \ H, 1 ^ ? sup |h, I otherwise h i * H S H A . 1 (x,h.)jt0 Lemma 6.4 i s the e q u i v a l e n t , i n V-spaces, of a theorem of C„ A. Swanson, v a l i d f o r H i l b e r t spaces: Theorem 1 of [ 3 4 ] * Theorem 2 of [ 3 5 ] » Th i s lemma i s used to prove the f o l l o w i n g comparison theorem: Theorem 6 . 5 - Let B £ ^ ( X ) and suppose that A, i s an eigenvalue of B, with the a s s o c i a t e d eigenspace Y^. If |B - A| < 1, then: 99 ( i ) A, i s a l s o an e i g e n v a l u e of A, ( i i ) t h e d i m e n s i o n of i s l e s s than or e q u a l to the d i m e n s i o n of P r o o f : Let H-^  be a d i s t i n g u i s h e d b a s i s f o r Y^. By Lemma 6 . 4 ? we have f o r each h 9 g H^s |h» - P A h « | = |Ah» - AJi»| - |Ah» - B h « | < | h « | . T h e r e f o r e . Hence, X^ i s n o n - t r i v i a l and ( i ) i s p r o v e d . By Theorem 2 = 2 .. 4 ( P a l e y - W i e n e r Theorem), t h e s e t P^H 9 i s a d i s t i n g u i s h e d s u b s e t of X^ and, by C o r o l l a r y 2-2.3 <? t h e c a r d i -n a l i t y of a d i s t i n g u i s h e d b a s i s of X^ i s g r e a t e r t h a n or e q u a l t o t h e c a r d i n a l i t y of P-^ H * „ ( i i ) f o l l o w s . The f o l l o w i n g c o r o l l a r i e s are i m m e d i a t e : C o r o l l a r y 6 . 6 . I f b o t h A and B admit c o m p l e t e s p e c t r a l decomposi-t i o n s and |B - A | < 1 , t h e n ( i ) A and B have t h e same e i g e n v a l u e s , ( i i ) f o r each e i g e n v a l u e A., t h e a s s o c i a t e d e i g e n s p a c e s f o r A and f o r B have t h e same d i m e n s i o n s . C o r o l l a r y 6 . 7 ° Suppose t h a t A admits a c o m p l e t e s p e c t r a l d e c o m p o s i t i o n and |-B - A § < 1 . I f A, i s an e i g e n v a l u e of A but i s not an e i g e n v a l u e of B, t h e n B does not admit a c o m p l e t e s p e c t r a l d e c o m p o s i t i o n . 100 Example 1. Thi s example shows that the converse of Theorem 6 - 5 ( i ) i s not t r u e , i . e . i f A has a complete s p e c t r a l decomposi-t i o n and |B - A| < 1, an eigenvalue of A i s not n e c e s s a r i l y an eigenvalue of B. Let H = {h^ s i = 0, 1, 2, • •> . } be a d i s t i n g u i s h e d b a s i s of a V-space X, with | h ^ | > | h ^ + ^ | f o r a l l i > 0. Define A and B by t h e i r values on H (Theorem Ah^ = h i i f i i s even, Aiu = 0 i f i i s odd, Bh i • h i i f i i s even, Bh i = h i + 2 i f ± ± a ^ A admits a complete s p e c t r a l decomposition and, by Theorem 6.2;, i t s only eigenvalues are 0 and 1. Let x £ X. Then f o r some N = N(x)s X = £ ( C X i h 2 i + P i h 2 i + 1 ) ' a i € F f f M F < * °» i>N l x I > l H 2H« 2 l ' A x = aN h2N + J a i h 2 i ' i>N+i Bx - a N h 2 N • I a i h 2 i * I hh2i*3' i>N +.1 i>N |Ax - Bx| - J £ p y i 2 1 + 3 | < l h 2 N + 2 & ° i>N Thus, |A - B| < 1. In accordance with Theorem 6 o 5 ( i ) * the eigenvalue 1 of B i s a l s o an eigenvalue of A. It i s e a s i l y v e r i f i e d that the eigen-value 0 of A i s not an eigenvalue of 8. It f o l l o w s from 101 C o r o l l a r y 6.7 that B does not have a complete s p e c t r a l decomposi-t i o n . Example 2. The f o l l o w i n g are l i n e a r operators on the space & ^ of 3 - 4 , f o r k > 0 . L CO ( i ) (£x)0O- = £ J e S c C O d t . o «>£ admits the complete s p e c t r a l decomposition i ( n 2 , ( p n ) l n = k, k + 1, k+2, ° ' ° } s i n c e cpn = n°(p n" <£ has a r b i t r a r i l y large eigenvalues and was s t u d i e d by T. E. H u l l [ 1 5 ] . Compare Jt with the Laplace Transform ( [ 8 ] , V o l . I ) : x(A.) -> f •e~ t x(X.)dX,o J o ( i i ) For \L > 0 , 0 0 - f V - O ^ - ^ d t . 7?1 admits the complete s p e c t r a l decomposition j i i d ) ) , n ^ i ) ) . „ k k + 1 k + 2 ... i [ V r ( ^ + n + l ) ' ?n/ ' n ' ' * ]• Comp are with the Riemann-Liouvi 1le f r a c t i o n a l i n t e g r a l ( [ 8 ] , V o l . I I ) : -1 _ t x (A.) - [ fao] ' J (t - A.)tJ-1x(A,)dA,. 102 ( i i i ) For K > 1, = r X JO(X - 1 ) d t . o t» admits the complete s p e c t r a l decomposition { (™, J^) s n = K, K + l , K+2, « ° • ] . Concerning t h i s c o n v o l u t i o n product, see, f o r example, M i k u s i n s k i [ 2 3 ] , pp. 174-178 and p. 456. Since none of the above operators £. , has eigen-value 0, they are i s o m e t r i e s : |<£x| = l ^ x l = I ^ x l = l X I F O R A 1 1 X € From Theorem 6.2, they have no other eigenvalues than those given i n t h e i r r e s p e c t i v e s p e c t r a l decompositions above. 4-7 Note on p r o j e c t i o n s A. F. Monna [24]« [25] has int r o d u c e d a n o t i o n of p r o j e c -t i o n i n non-Archimedean normed l i n e a r spaces. In the s p e c i a l case of V-spaces we have the f o l l o w i n g : Def i n i t i on 7.1. Let Y be a c l o s e d subspace of a V-space X. An operator P £ ^ ( X ) i s c a l l e d a p r o j e c t i o n on Y i f f o r a l l x f X, Px £ Y and (4.16) |x - Px| < f x - y j f o r a l l y g Y. Theorems on p r o j e c t i o n s and comparisons with p r o j e c t i o n s i n H i l b e r t space theory [ 7 ] , [ 3 6 ] w i l l be found i n Monna [ 2 4 ] , 103 Part IV and [ 2 5 ] , Part I. The e x i s t e n c e and non-uniqueness of p r o j e c t i o n s on a given subspace Y of X were proved by Monna. The proofs of Monna do not i n v o l v e e x p l i c i t l y the use of d i s t i n g u i s h e d bases. We give here an a l t e r n a t e and simple pro o f . Let H(Y) be a d i s t i n g u i s h e d b a s i s of Y and H be an a r b i -t r a r y extension of H(Y) to a l l of X. Denote by Z the c l o s e d subspace generated by H N H ( Y ) . Define the l i n e a r operator P on X by i t s values on H (Theorem 4.1) s Ph H h i f h £ H(Y), Ph = 0 i f h £ H S H ( Y ) . By Theorem 2-4=4 and C o r o l l a r y 2-3»4, the spaces Y and Z are d i s t i n g u i s h e d complements of one another. Therefore f o r each x £ X, there e x i s t y £ Y and z c Z such that x a y + z . Since X *2* X X the r e s t r i c t i o n of P to Y i s the i d e n t i t y mapping and i t s r e s t r i c -t i o n to Z i s the 0-operator: Px = Py +Pz = y £ Y, and f x - Px| « f x - y x ) = | z x | -For an a r b i t r a r y y £ Y„ y - y £ Y and, since Y and Z are d i s t i n g u i s h e d subsets of X; f x - y | = | ( y x - y) + zJ « Max[8y - y j , |z x|} Hence (4.16) i s s a t i s f i e d . This proves that P i s a p r o j e c t i o n on Y. 104 The non-uniqueness of the p r o j e c t i o n s on Y i s a con-sequence of the non-uniqueness of the extensions H of H(Y). Remark: The operator of Lemma 6.4 i s a p r o j e c t i o n on X^ (see page 98). 105/ CHAPTER 5 S o l u t i o n of E q u a t i o n s 5 - 1 I n t r o d u c t i on The p r o b l e m s t u d i e d by C. A„ Swanson and M. S c h u l z e r i n [ 3 2 ] and [ 3 3 ] i s t h a t of t h e e x i s t e n c e and t h e a p p r o x i m a t i o n of a c l a s s of e q u a t i o n s i n Banach s p a c e s . In t h i s C h a p t e r we g e n e r a l i z e Theorems 4 and 5 of [ 3 3 ] t o a r b i t r a r y V - a l g e b r a s and V - s p a c e s . The h y p o t h e s e s of [ 3 3 ] are s l i g h t l y weakened. 5-2 E q u a t i o n s i n V - a l g e b r a s In t h i s S e c t i o n , X i s a V - a l g e b r a . We c o n s i d e r two p o i n t s , x, y £ X which have t h e f o l l o w i n g f i n i t e or i n f i n i t e e x p a n s i o n s : o 1 2 y = y o + y l + y 2 + ""' and we assume t h a t x admits a p s e u d o - i n v e r s e x 1 such t h a t o o ( 5 . 1 ) f x - X Q § < I x - 1 ! " 1 . I t f o l l o w s f r o m Theorem 2 - 6 . 6 t h a t : The orem 2 . 1 . The element x a d m i t s a p s e u d o - i n v e r s e x 1 and t h e e q u a t i o n xw = y a d m i t s a p s e u d o - s o l u t i o n z = x ( i . e . xz = y ) . The p r o b l e m i s t o make use of t h e known e x p a n s i o n s of x and y t o o b t a i n a p p r o x i m a t i o n s t o z and x as d e f i n e d i n t h e above 106 t h e o r e m . The s e q u e n c e s [z } a n d f u } d e f i n e d b y n n ri ' n ( 5 . 2 ) z = x 1 y „ z = x 1 [ Y y , - Y x , z ..) , v 0 0 o n 0 V Z, 1 Z . 1 n - i / i = 0 i = l n (5.3) u = x 1 , u = x 1 ( e - Y x, u . ) , i = l - 1 w i l l be shown t o a p p r o x i m a t e z a n d x , r e s p e c t i v e l y , p r o v i d e d t h e r a t e s of c o n v e r g e n c e of t h e s e r i e s x^ a n d y n s a t i s f y c e r t a i n c o n d i t i o n s . M o r e p r e c i s e l y , we s h a l l c o n s i d e r t w o s e t s o f a s s u m p t i o n s o n t h e r a t e s o f c o n v e r g e n c e o f t h e s e r i e s ^ x^ a n d ^ y ^ a n d , u n d e r t h e s e a s s u m p t i o n s we s h a l l o b t a i n u p p e r b o u n d s f o r t h e • v a l u e s o f | z - z n | and | x 1 - U n B ' ° I n t h e f i r s t c a s e we assume t h a t (5.4a) L X N I < P " 1 1 ! ^ 1 ! ! " 1 F O R N ^ 1> ( 5 . 4 b ) I y n 1 < ^ ~ n | y 0 1 f o r n > 1 . I n t h e s e c o n d c a s e , o u r a s s u m p t i o n s a r e t h a t ( 5 . 5 a ) | x j > f x j > | x 2 | > , ( 5 . 5 b ) § x^§° | x m I < f x m + n l ° S x 0 " ^ I ^ f o r a l l n , m > 1 s u c h t h a t ' f x , | * 0 , 1 m+n • ' ( 5 . 5 c ) J y o | > | y n | f o r a l l n > 1 , ( 5 . 5 d ) | y I < | x n _1|.|x"1|.Max{|x | f | y 1} f o r a l l n > 1. Il it ™ J. o o o 1 0 7 The i n t e r e s t of the second case l i e s i n i t s a p p l i c a b i l i t y in V-algebras which admit d i s t i n g u i s h e d bases with many elements having the same norm (e.g. the V-algebra of 3 -5)» In such cases, the norms of the terms i n the expansions of x or y w i l l not n e c e s s a r i l y decrease as r a p i d l y as r e q u i r e d by ( 5 » 4 ) , and to sum up the terms having the same norms may be inconvenient or d i ff i c u I t . Theorem 2.2. (i) If ( 5 . 4 a ) and ( i>»4b) h o l d , then the sequence [z } d e f i n e d by (5.2) converges to z and n > (5.6) |z - z n | < p ~ n L ^ " 1 ! fy o | f o r a l l n = 0, 1, 2, ° • - . ( i i ) If ( 5 . 5 a ) , ( 5 » 5 b ) , (5.5c) and (5°5d) h o l d , then 1 2 ( 5 . 7 ) |z - z n | < l x n | | x ~ I Max(Jx Q|, [y Q|} f o r a l l n such that | x | * 0 ; * n • if f o r a l l i n t e g e r s n, | x n | ^ 0» then [ z n l converges to z. Before p r o v i n g the theorem, we note that i f y Q s e and y n = 0 f o r a l l n > 1, then ( 5 . 4 b ) , (5.5c) and (5»5d) ,are s a t i s -f i e d and, hence, the f o l l o w i n g c o r o l l a r y i s deduced from Theorem 2.2: C o r o l l a r y 2.3. ( i ) If ( 5 . 4 a ) h o l d s , then the sequence {u^l d e f i n e d by (5=3) converges to x 1 and I - " 1 " % ! < f ^ l ^ l f o r a 1 1 n = 0, 1, 2 , " — . 108 ( i i ) If (5-5a) and (5.5b) h o l d , then I * " 1 " u n | < g x j j x ; 1 ! 2 M a x d x J , 1} f o r a l l n such that Ix I * 0 ; • n • i f f o r a l l i n t e g e r s n, J x n | £ 0, then ( u n J converges to x ^, Proof of Theorem 2.2. One v e r i f i e s d i r e c t l y that z = x o 1 [ y - ( x - x o ) z ] = x ~ X £ £ y . - ( £ x . ) zj . i>0 i > l Thus, ' K * ( J - y i ) " ( I X i ) z -If i>n+l i > n + i n - x ; x ( I x i ( z - z n - i } ) I i = l and (5.8) l z - z^ I < Max[a , (3 y n K where a n " K ^ ' l I * ± l i >h +1 i>n+l n i = l Both (5.4b) and (5.5c) imply | y | < j y o f ; hence, from Theorem 2 - 6 . 6 ( i i ) and the r e l a t i o n 1 < f x 0 i i x 0 8 ° (5.9) l » l < h " 4 l y | < l ^ l l y o l . 1 0 9 This shows that both ( 5 - 6 ) and ( 5 = 7 ) are s a t i s f i e d f o r ri' = 0 . We complete the proof by induction., and f o r each set of assump-t i o n s s e p a r a t e l y . ( i ) In the f i r s t case, suppose that ( 5 ° 6 ) i s s a t i s f i e d f o r n = 0, 1, 2, •'°, m-1. From (5.4b) and Theorem l - 4 . 2 ( i i ) : from (5.4a), Theorem l - 4 . 2 ( i i ) and (5-9) • ^ < K ' l K . J M S / ' - ' K ' l l y . l . . from (5.2a) and the i n d u c t i o n h y p o t h e s i s : Y < I x " 1 ! Max {|x,|lz-z .1} < 0 " m | x " 1 | | y I ' m — 1 o • _. . 1 • 1 " • m-i 1 — P • o 1 • J o • l<i<m / It f o l l o w s from ( 5 . 8 ) and these three i n e q u a l i t i e s that ( 5 . 6 ) holds f o r n = m and hence f o r a l l n. The convergence of p n to 0 i m p l i e s that l i m ] z - z | = 0 and, consequently, l z n l converges to z. ( i i ) In the second case, we note that (5.5a) i m p l i e s that when | x m I 4= 0, then | x^ | i 0 f o r n = 0, 1, • ° » , m-1. Suppose that ( 5 . 7 ) holds f o r n = 0, 1, °°°, m-1. Then an argument s i m i l a r to that conducted i n the f i r s t case shows that ( 5 - 7 ) holds also f o r n = m. If f o r each i n t e g e r n, f x n | ^ ®« the convergence of the s e r i e s x n i m p l i e s , as i n the f i r s t case, the convergence of (z } t o z. 110 A p p l i cat i on. Let Z be an a r b i t r a r y V-space. Let {A^ : n = 0, 1, 2, •••} be a sequence of l i n e a r operators i n the V - a l g e b r a IT (Z) = Assume th a t , i n the norm of !J(Z), A = A + A. + A_ + ••• . o 1 2 Assume a l s o that A q i s pseudo-regular, with pseudo-inverse A and that o (5.10) |A - A O | < I A ; 1 ! - 1 . (5.1D K i < P ^ I A ; 1 ! " 1 . Under these assumptions the equation (5 . 12) Az = w, z, w € Z has a s o l u t i o n z f o r each w £ Z. Indeed, from C o r o l l a r y 2.3, A has a pseudo-inverse A \ so that z = A w i s a s o l u t i o n of ( 5 . 1 2 ) . Furthermore, i t f o l l o w s from (5«3) that A ^ i s a l i m i t of the sequence [B }: n n B = A - 1 , B = A" 1 ( I - V A.B .) o o n o * ZJ 1 N _ 1 ' i = l and, by the l i n e a r i t y of the operators A \ z i s a l i m i t of the sequence [z j : n n z = A "^w, w = B w = A ^ (w - y A.w .} . o o ' n n o \ , £J 1 N _ 1 ' i =1 I l l (Compare t h i s r e s u l t with the r e s u l t s of S e c t i o n 4 below.) 5 - 3 The equation Ax = y In t h i s s e c t i o n , X and Y are V-spaces, A £ Qf(Z,. Y). D e f i n i t i o n 3 . 1 . Let y £ Y and D c X. ( i ) The equation Ax = y i s s a i d to have the pseudo- s o l u t i on z i n D i f z £_ D and Az = y. ( i i ) The equation Ax « y i s s a i d to have a unique pseudo- s o l u t i on i n D i f i t has at l e a s t one p s e u d o - s o l u t i o n z i n D and i f z' = z f o r a l l p s e u d o - s o l u t i o n s i n D. We c o n s i d e r the l i n e a r operator A^ £ 7 ( X , Y) and assume that A has a bounded pseudo-inverse A ^ on i t s range A(X). The O O O operator A ^ i s l i n e a r . (See Theorem 4 ~ 3 ° 4 ) ° o Theorem 3 . 2 . Let y Q € A Q ( X ) a n c * u = A 0 l y o ° I f t h e r e e x i s t s a b a l l D = S»(u, r ) , r > 0 , such that K = J A " 1 ^ ^ and such that the f o l l o w i n g c o n d i t i o n s ( 5 . 1 3 ) and ( 5 = 1 4 ) are s a t i s f i e d ? ( 5 . 1 3 ) |A. - A o f i D < K" 1; ( 5 . 1 4 ) (A - A Q ) X f S»(0, r K " 1 ) d A Q ( X ) f o r a l l x £ D; then, f o r a l l y £ A Q ( D ) the equation Ax = y has a unique pseudo-s olu t i on z i n D. Furthermore, the sequence {z } d e f i n e d by (5 = 1 5 ) z Q - A ; X y , z n = A ^ y - A ^ ( A - A ^ z ^ , converges to z. 112 Proof : Let y» = y - y Q . Since y £ a q ( D ) , A ~ Y € D and (5.16) l A ^ y * | = | A ^ y - u | < r . S i n c e , from ( 5 . 1 4 ) , ( A - A Q ) X £ A ( X ) f o r a l l x € D, the equation ( 5 . 1 7 ) Ax = y - y Q + y' i s e q u i v a l e n t , f o r x £ D, to the equation (5.18) Lx = x, where Lx = u + A~ 1y s' - A _ 1 ( A - A ) X , X € D. o o o ' From ( 5 = 1 4 ) | A _ 1 ( A - A ) x | < K . r K - 1 = r f o r a l l x € D. • o o 0 — From t h i s i n e q u a l i t y and ( 5.16), i t f o l l o w s that |Lx - u | - j A ^ y * - A ~ 1 ( A - A Q ) x j < r f o r a l l x € D. Thus, L maps D i n t o i t s e l f . From ( 5 . 1 3 ) , we have, f o r a l l x^, x^ € D: | L x x - L x 2 J = | A ; X ( A - AQ)X1 - A ^ ( A - A o ) x 2 | < K.|A - A o | D J x 1 - x 2 | < |x 1. - x 2 J . Since 0 i s the only accumulation p o i n t of the norm range of a V-space, i t f o l l o w s that l - | D < f 1 < 1-The c o n t r a c t i o n mapping p r i n c i p l e ( [ 1 9 ] , V o l . I, p. 4 3 ) can 1 1 3 be a p p l i e d to L on the c l o s e d sphere D, to conclude that the equation (5 = 1 7 ) and, hence, the equation (5 = 18) have a unique p s e u d o - s o l u t i o n z i n Do The c o n t r a c t i o n mapping p r i n c i p l e a l s o a s s e r t s that the sequence l z n l d e f i n e d by z • = A "*"y, z' = Lz , o o J ' n n - 1 converges to the p s e u d o - s o l u t i o n z. S i n c e K - | A ; X | A ( D ) < J A ^ ^ ( x ) and \ A - A j d < j A - A j J t o o we see that the theorem holds i f , i n ( 5 = 1 3 ) , |A - A o J ^ i s r e -p l a c e d by JA - A o|^ and/or i f , i n one or both of ( 5 = 1 3 ) and (5»14K K i s r e p l a c e d by |A (X) ° o T h i s theorem extends Theorem 4 of [ 3 3 ] (Th. 7 . 1 of [ 3 2 ] ) a r b i t r a r y V-spaces. A p p l i c at i on. For some i n t e g e r k (> 1 , l e t X = Y = where (Pk i s d e f i n e d i n S e c t i o n 3 - 4 = We c o n s i d e r an operator F £ ©"(tf^) such that (5 = 1 9 ) 0 < | I - F I < 1 . (Examples of such operators are F = F^ where F^x = x + x n, n = 2 , 3 , o r F^x = x ( l + «pn ) * n = 1 , 2 , = = = . ) Let S£ be the operator d e f i n e d i n S e c t i o n 4 _ 6 , page 1 0 1 ; n amely : 0 0 1 ~7x ( X x ) W - J l ^e * x ( t ) d t . 114 Consider the equation ( 5 . 2 0 ) y + £ F x = ax, t. ( i . e . : y(A.) + ^e F ( x ( t ) ) dt = ax (A.) ), where y € (P^ a ^ d a o i s a r e a l number. We s h a l l apply Theorem 3=2 to prove that ( 5 . 2 0 ) has a unique p s e u d o - s o l u t i o n i n (P^ when ( 5 . 2 1 ) a * nl f o r each i n t e g e r n > k. Define, f o r x £ (P^ 1 Ax = ax - ^£.Fx « ( a l - aC.F)x A Q X = ax - £. x = ( a l - £ ) x The equation ( 5 . 2 0 ) i s eq u i v a l e n t to the equation ( 5 . 2 2 ) Ax = y It f o l l o w s from the r e s u l t s of page 101, 4 - 6 , that i f a * n l f o r each i n t e g e r n > k, A q = a l - £ i s pseudo-regular and that i t s pseudo-inverse A ^ i s d e f i n e d on a l l of (P. , with o k -k To apply Theorem 3 - 2 , s e l e c t JQ = u = 0 and r = p Then, D = (P. . k Since A - A = £ ( I - F) and f & | = 1, we have from ( 5 . 1 9 ) |A - A J < - P | < 1 - I A ; 1 ! " 1 , and hence, ( 5 » 1 3 ) i s s a t i s f i e d . 1 1 5 C l e a r l y ( 5 - 1 4 ) i s also s a t i s f i e d since ( A - A Q ) maps (?^ i n t o i t s e l f and si n c e |A^ "^  | = 1 . The c o n c l u s i o n i s that ( 5 = 2 2 ) and ( 5 - 2 0 ) have a unique p s e u d o - s o l u t i o n z i n (P ^  when ( 5 = 2 1 ) h o l d s . Furthermore, z i s a l i m i t of the sequence i z n l d e f i n e d by z - A _ 1 y , z = A " 1 y - A - 1 SC. ( I - F)z o o J ' n o o n - 1 Other examples of a p p l i c a t i o n s of Theorem 3 - 2 w i l l be found i n [ 3 2 ] . 5 - 4 The equation Ax = y i n v o l v i n g expansions of A and y. As i n the pr e v i o u s S e c t i o n , X and Y are V-spaces, A £ QT(X, Y) and we con s i d e r the equation Ax = y. However, we now suppose that A and y are known from t h e i r f i n i t e or i n f i n i t e expansions a n A = A + A. + A„ + o 1 2 y = y Q + + y 2 + * * " We assume that A n ^ 0~(X, Y) f o r n = 1 , 2 , that A c *3 (X, Y) and that y £ A (X) . We al s o assume that A has o o ^ o o a pseudo-inverse A on i t s range A (X) . Let u = A "^ y . ~ o ^ o v o o n Suppose that there e x i s t s a b a l l D = S'(u, r ) , r > 0 , such that K = J A ^ I ^ and such that the f o l l o w i n g c o n d i t i o n s (5 . 2 3 ) - (5 - 2 6 ) are s a t i s f i e d : ( 5 . 2 3 ) IA - A o f D < K""1; ( 5 . 2 4 ) j A n x g < r K - 1 m i n [ l , S A n S D l f o r a 1 1 n > 1 a n d a 1 1 x € D; 116 (5.25) | y n | < r K " 1 f o r a l l n > 1; (5.26) In Y, the b a l l S»(0, r K - 1 ) ' i s contained i n A (X) . Theorem 4 » 1 • Under the c o n d i t i o n s above, the equation Ax = y has a unique p s e u d o - s o l u t i o n z i n B. Proof ; The convergence on D of the s e r i e s ^ n > Q A^ i m p l i e s that l i m | A n 8 D = 0. Therefore, from ( 5 . 2 4 ) , l i m |A *| = 0 f o r n«*» n«*os a l l x £ D and, hence, the s e r i e s ^ A^x i s convergent on D. Then, i t f o l l o w s , a l s o from ( 5 . 2 4 ) that (5 .27) F ( A - A q ) X | = G £ A n x B < r K - 1 f o r a 1 1 x D ° n>l A consequence of (5.25) and (5.26) i s that y e A (X) f o r n ^ o a l l n > 1 and that From the l i n e a r i t y of A q we conclude that y £ A Q ( X ) and the l a s t i n e q u a l i t y g i v e s n>l Hence s (5.28) y £ A (D). o The r e l a t i o n s ( 5 . 2 3 ' ) , (5 .24), (5 .27) and (5.28) e s t a b l i s h the a p p l i c a b i l i t y of Theorem 3 » 2 . Thus, Ax = y has a unique p s e u d o - s o l u t i o n z i n D. 1 1 7 As i n Theorem 2 . 2 , we now seek an approximation to the p s e u d o - s o l u t i o n z. We c o n s i d e r the sequence {z } d e f i n e d by n n ( 5 . 2 9 ) z = A - 1 y , z = A" 1 ( V y, - V A.z . "\ . o o-'o' n o \ £ J 1 L 1 n - i / i=0 i = l The e x i s t e n c e of t h i s sequence i s guaranteed by the f o l l o w i n g 1emm a . 1 n n Lemma 4 . 2 . Let u = ) y. - ) A,z . , n = 1 , 2 , - n Z-i 1 Lx 1 n - i ' ' ' i=0 i = l The domain of A ^ c o n t a i n s a l l u , n = 1, 2 . ••• and z £ D o n n v f o r a l l n =0, 1 , 2 , • • • . Proof ; C l e a r l y Z Q £ Dp Suppose that z^ £ D f o r i = 0, 1 , 2 , n - 1 . Then from ( 5 . 2 4 ) . - 1 I A i Z n - i I - r K f o r i = 1* 2 , n, and ( 5 . 2 6 ) i m p l i e s that A.z . 5 A ( X ) f o r i = 1 , 2 , n. 1 n - i o . It was j u s t shown that y^ £ A Q ( X ) f o r a l l i > 0. Hence, u. c A ( X ) f o r i = n . T h i s i n d u c t i o n shows that u. £ A ( X ) 1 ^ 0 1 0 f o r a l l i > 1 , provided z^ ^ D f o r a l l i > 0. By i n d u c t i o n , z^ £ D f o r a l l i > 0, s i n c e by ( 5 . 2 5 ) and the above i n e q u a l i t y ; n i = l In Theorem 4 ° 3 we show that z^ i s an approximation to z and we give an upper bound f o r |z - z^$. As i n Theorem 2 . 2 , the degree of t h i s approximation depends on the rates of convergence 118 of the s e r i e s ^ n > g A^ and ^  y^. In that r e s p e c t , -we make two d i s t i n c t sets of a d d i t i o n a l assumptions on A^ and y n . F i r s t : (5.30a) | A N J D < p " n K _ 1 f o r n > 1, . ( 5 = 3 0 b ) | y n | < T p ~ n * 1 j T 1 f o r n > 1; secondly : ( 5 . 3 1 a ) G A O L D > | A ; L | D > I A 2 | D > , ( 5 . 3 1 b ) | A M I D L A N 8 D < K " 1 | A M + N L D f o r a l l m > 1 and n > 1 such that | A m + n | * 0 , ( 5 . 3 1 c ) | y 0 | > | y n | f o r a l l n > 1, ( 5 - 3 l d ) | y n | < R K " 1 m i n [ l , |A xlj,} f ° r a 1 1 n > 1. Assumptions (5.30'b) and (5»3ld) imply (5.25). •1, Theorem 4 . 3 . ( i ) If ( 5 . 3 0 a ) and ( 5 . 3 0 b ) h o l d , then the sequence f z n l d e f i n e d by ( 5 . 2 9 ) converges to z and ( 5 . 3 2 ) | z - z n J < r y ~ n Max[l, K" 1} f o r n = 0, 1, 2, . ( i i ) If (5.31a), (5.31b), ( 5 . 3 1 c ) and (5-3ld) h o l d , then (5.33) J z - Z N | < R L A N J D M a x i 1 ' K _ : L 5 f ° r a l l ,n such that K I D * 0 ; i f f o r each i n t e g e r n > 0, § A ^ 0, then the sequence f z } "~ • n • 1) n converges to z. 1 1 9 Proof: From (5.27), i t f o l l o w s that (A - A )z c A (X). Hence [y - (A - A q ) Z ] belongs to the domain of A" 1. If may be v e r i f i e d d i r e c t l y that ( 5 . 3 4 ) z - A ; 2 [ y - (A - A Q ) z ] - A ^ ( £ y n - £ \ * ) • n>0 n>l From the d e f i n i t i o n of { z n l and the l i n e a r i t y of A" 1, we have, f o r n = 1 , 2 , 3 , • • • I * - •»l-K l(I"'i) - ^ o 1 (. I v) i>n+l i>n+l n i = l Hence, - z N L < Max[a n, 0 n, y n} f o r n = 1 , 2 , where a n i>ri +1 i>n + l and, s i n c e z £ D and z^ £ D f o r a l l i > 0 (Lemma 4 . 2 ) , ^n " KV*" H A i l D ° f i Z ~ Z n - i l ^ l<i<n S i n c e 1 < |A 0| D-|A; 1| D # I z ~ z 0 i < r < r Max{l, K - 1 } , l z ~ z 0 | < r < r | A o 8 D Max{l, K - 1} 120 So, ( 5 . 3 2 ) and ( 5 - 3 3 ) are both s a t i s f i e d f o r n = 0. The r e s t of the proof i s conducted, f o r each set of assump-t i o n s ( 5 . 3 O ) and (5 • 31) # by i n d u c t i o n , and e x a c t l y as i n Theorem 2 . 2 . We omit t h i s l a t e r part of the p r o o f . It i s e a s i l y v e r i f i e d that Theorems 4 ° 1 » 4 - 3 and Lemma 4 . 2 h o l d i f i n the hypotheses ( 5 - 2 3 ) , ( 5 . 2 4 ) , ( 5 » 3 0 a ) , ( 5 . 3 1 a ) , (5.3lb), (5.3ld) and the estimate (5-33), we change a l l norms on D (JAn8D) to norms on X (IAnix) -If we assume that X = Y and that a l l operators A^ are l i n e a r , and that the above change to norms on X i s made, the r e s u l t s of Theorem 4 - 1 and 4 - 3 are refinements of those of the A p p l i c a t i o n of Theorem 2 . 2 , page 110. A p p l i c a t i o n s can be found i n [ 3 2 ] . 1 2 1 CHAPTER 6 CONTINUOUS LINEAR FUNCTIONALS 6-1 Dual space In t h i s Chapter, X i s a V-space over the f i e l d of s c a l a r s F. F i s a V-space over i t s e l f and i s given the d i s c r e t e topology induced by i t s t r i v i a l v a l u a t i o n . * The term A f u n c t i o n a l on X,B w i l l be used to denote an operator from X to F. D e f i n i t i o n 1.1. The space X* = 17(X, F) of bounded l i n e a r f u n c t i o n a l s on X i s c a l l e d the dual space of X. Theorem 1 . 2 . ( i ) X* i s a V-space. ( i i ) Every continuous l i n e a r f u n c t i o n a l on X i s bounded and belongs to X*. ( i i i ) For each x f X and f £ X* there e x i s t s r > 0 such that f (S (x, r) ) = f (x) . Proof : ( i ) and ( i i ) are s p e c i a l cases of Theorem 4 _ 3 ° 1 and Theorem 4 - 3 ° 5 ? r e s p e c t i v e l y . ( i i i ) f o l l o w s from the c o n t i n u i t y of f and the d i s c r e t e n e s s of F. A d i r e c t proof of the v a l i d i t y of the Hahn-Banach Theorem (Th. 1.3 ( i ) ' below; [ 3 6 ] , p. 186) i n V-spaces has been given by A. F. Monna ([24], Part I I I , pp. H 3 7 - H 3 8 ) . A. W. Ingleton Since [0] = [o}, 'the symbols " s l t and n = M have the same meaning i n the V-space F. (See page 4 5 = ) 1 2 2 [ 1 7 ] c o n s t r u c t e d a proof based on the notion of s p h e r i c a l com-pl e t e n e s s (see 2 - 5 , ( i i ) ) . Another proof i s due to I. S. Cohen [ 1 3 ] , P- 6 9 6 . Monna has a l s o proved (same r e f e r e n c e ) the e x i s -tence of the l i n e a r f u n c t i o n a l s r e f e r r e d to i n ( i i ) of the f o l l o w i n g theorem. Theorem 1 . 3 . ( i ) Let Z be a subspace of X. To each l i n e a r f u n c t i o n a l f ^ £ Z* there corresponds at l e a s t one l i n e a r func-t i o n a l f ^ ^ X* such that ( 6 . 1 ) l f 2 ^ X = i f l B z S n d f 2 ^ 5 f n i * } f o r a l l x f Z. ( i i ) For X q g H, J X Q | * 0 and every s c a l a r a £ F, a f 0, there e x i s t s f £ X* such that f ( x Q ) s a and | f | = J x o | - 1 . Proof ; See the r e f e r e n c e s quoted above. We gi v e a new proof of ( i ) , u s i n g Theorem 4 - 4 » I - Let H v be a d i s t i n g u i s h e d b a s i s of Z and H be an a r b i t r a r y extension of H» to a l l of X (see D e f i n i t i o n 2 - 4 . 3 ) . On H, d e f i n e f 1 ( h ) f o r h € H» 0 f o r h ^ H\H' f 2 ( h ) - i It f o l l o w s from Theorem 4 ~ 4 ° 1 that i s determined on X by i t s values on H and that ( 6 . 1 ) i s s a t i s f i e d . * To prove ( i i ) , d e f i n e f ^ ( x . Q ) 5 a and extend f ^ by l i n e a r i t y to the subspace [x ] . Then | f , I r 1 = |x 8 ^. The c o n c l u s i o n o • 1 • I x I o • f o l l o w s from ( i ) . 123 Theorem 1 .4- One of X and X* i s a bounded space i f and only i f the other one i s a d i s c r e t e space. P r o o f : It f o l l o w s from Theorem l - 3 ( i i ) that i f X i s not d i s -c r e t e , i . e . i f there are p o i n t s i n X with a r b i t r a r i l y small non-zero norms, then X* i s unbounded. The same theorem im-p l i e s that i f X i s unbounded there e x i s t l i n e a r f u n c t i o n a l s of a r b i t r a r i l y s m a l l non-zero norms. Suppose that X* i s unbounded. Then, f o r any i n t e g e r K > 0 there e x i s t s f g X* with | f J > K. Since there must ,be a point x £ X f o r which | f (x) | = 1 = | f || x |, there must be n o n - t r i v i a l p o i n t s of X with norms l e s s than K ^. Hence, X i s not d i s c r e t e . F i n a l l y , suppose that X i s bounded, i . e . f o r some M > 0 , | x | < M < f o r a l l x £ X. For a l l f £ X*, | f | =^  0 , we have | f ( x ) | = 1 < | f |" f x | f o r a l l x such that f (x) £ 0 . Thus, | f | > and X* i s d i s c r e t e . 6-2 The * norm on (H) Let H = [h^ : i £ j } , where J i s some index set, be a d i s t i n g u i s h e d b a s i s of X. (H) denotes the set of a l l f i n i t e l i n e a r combinations of elements of H. In t h i s s e c t i o n we s h a l l d e f i n e a new norm on the elements of (H). In the next s e c t i o n we s h a l l use t h i s new norm to e s t a b l i s h the r e l a t i o n s h i p between X* and (H). 124 The symbol w ( x , h)|f" was i n t r o d u c e d on page 28. D e f i n i t i o n 2.1. For x £ X, ( i ) J ( x ) = [ i e J : (x> * . ) H * 0}; ( i i ) HJ(X) l s d e f i n e d by the r e l a t i o n : | x | = p tt,^x^; ( i i i ) = sup {(a(h )}, j ( 9 ) = (-»). i*J,U) For x £ X, J ( x ) i s countable and x = ^ (x, n ^ ) j j h ^ . i € J ( x ) For y ^ ( H ) , J ( y ) i s f i n i t e . It i s e a s i l y v e r i f i e d that f o r a l l y, z e ( H ) : ( i ) {(ay) = i(y) f o r a l l a £ F, a * 0; (6.2) ( i i ) i(x + z) j < Max U (y ) „ J (z) } . = Max U ( y ) , *(z)} whenever I (y) £ i ( z ) . The two sets of i n t e g e r s iaj(h^) : h^ £ H } and (^(h^) : h^ ^ H J are i d e n t i c a l since f o r each h^ £ H , 3,(1^) = l ( h i ) o The set [^(h^^) : h i ^ H} i s bounded above i f and only i f X i s a d i s c r e t e space; i t i s bounded below i f and only i f X i s bounded i n i t s norm. D e f i n i t i o n 2.2. The f u n c t i o n which assigns to each p o i n t y of ( H ) the non-negative r e a l number (6.3) |y|* - p * ( y ) w i l l be c a l l e d the * norm on ( H ) . Theorem 2.3 » ( i ) Under the *norm ( H ) has a l l the d e f i n i n g 1 2 5 p r o p e r t i e s of a V-space, except p o s s i b l y when X i s unbounded, i n which case (H) may not be complete. ( i i ) One of the spaces X and (H), under the *norm, i s bounded i f and only i f the other i s d i s c r e t e . ( i i i ) The set H i s a d i s t i n g u i s h e d Hamel b a s i s of (H) under the *norm. Proof; Except f o r the completeness requirement, ( i ) i s e a s i l y proved from ( 6 . 2 ) and ( 6 . 3 ) . : ( i i ) f o l l o w s from the remark pr e c e d i n g D e f i n i t i o n 2 . 2 , and the f a c t that the set ll(h^) : h^ £ H} i s bounded above i f and only i f (H) i s bounded under the *norm; - i s bounded below i f and only i f (H) i s a d i s c r e t e space under the *norm. If X i s bounded, the completeness of (H) f o l l o w s from i t s di s c r e t e n e s s . ( i i i ) f o l l o w s from the f a c t that f o r a l l y £ (H) such that fy 1* r- 0: Max l w ( h , ) ] ) | y | . . <(y) i € J ( y ) = M a x { - ( h i > } f ? i € J ( y ) p ) = Max [ p 1 } = Max {|h |*} i£j(y) r i € J ( y ) 6 - 3 H-inner product and r e p r e s e n t a t i o n theorems To the n o t a t i o n s , d e f i n i t i o n s and hypotheses of the pre v i o u s s e c t i o n , we add the assumption that the f i e l d of s c a l a r s , F, i s the f i e l d of the r e a l or complex numbers, a denotes the complex conjugate of a £ F. 126 D e f i n i t i o n 3.1. ( i ) j ( x , y) = j ( x ) n J (y) • ( i i ) The s c a l a r valued f u n c t i o n , d e f i n e d on X x (H) by 0 i f J(x, y) = 0 , <x, y> H = < I ( x , h . ) H - ( y , h . ) H i f J ( x , y) * 0 , i € J l x , y ) x £ X, y £ (H), w i l l be c a l l e d the H-inner product on X. The f o l l o w i n g p r o p e r t i e s of the H-inner product are e a s i l y v e r i f i e d : For a l l u, v £ X, a l l y, z £ (H) and a l l a, 8 £ F: <y, z > H = <z, y> H; <au, By> H = a8<u, y> H; <u+v, y + z > H = <u, y> H + <v, y> H + <u, z> f l + <v, z> H. The analogy with the u s u a l inner product i s evident ( [ 7 ] , p. 2 4 2 ; [ 1 9 ] , Part I I , p. 80; [36], p. 106). An important d i f -ference i s i t h a t the H-inner product depends on H. Indeed, given two d i s t i n c t d i s t i n g u i s h e d bases H.^  and H,, of X, i f y £ (H^) and y £ ( H 2 ) , then, f o r a l l x £ X, <x, y> H i s d e f i n e d but <x, y> H i s not; i f y £ (H^) f] C*^), then there may e x i s t x £ X such that <x, y>„ T4 <x, y>„ . To pursue the analogy, we s h a l l 1 2 e s t a b l i s h a r e l a t i o n s h i p between the H-inner product and the bounded l i n e a r f u n c t i o n a l s on X. An i s omorph i sm between two V-spaces X and Y i s a one-to-one continuous l i n e a r operator from a l l of X to a l l of Y. An i s o m e t r i c isomorphism y i s an isomorphism such that ;x)| = J x | f o r a l l x g X ( [ 7 ] , p. 65). 127 Theorem 3 * 2 . There e x i s t s an i s o m e t r i c isomorphism (p^ between (H) with i t s *norm and a subspace of X*; f o r a l l y g (H), <pH(y) s f y i s such that ( 6 . 4 ) f (x) = <x, y>„ f o r a l l x c X. y ri ^ Furthermore, the set epjj(H) i s a d i s t i n g u i s h e d Hamel b a s i s f o r the subspace -, H((H)) o f X*; f o r f £ <p H((H)), ( 6.5) f y . I (y, h . ) H f h i € J ( y ) in the norm of X*. Proof ; For each f i x e d y £ (H) i t i s easy to v e r i f y that the mapping d e f i n e d by ( 6 „ 4 ) i s a l i n e a r f u n c t i o n a l on X. Let ?H be the operator on (H) d e f i n e d by <pjj(y) s f y« >^JJ i s l i n e a r since the H-inner product i s l i n e a r i n y. a) If J ( x , y) = 0, then I f y(x)| = 0. b) If J ( x , y) * 0, then 0)(x) < 0 ) ( h i ) f o r a l l i £ J ( x ) , «j(h.) < {(y) f o r a l l i e J ( y ) . T h e r e f o r e , op(x) < 1 (y) and j f y ( x ) | - 1 < ^ ( y ) . p - » ( x ) - l y f * - ! * ! -c) Since l(y) i s f i n i t e , there e x i s t s i £ J such that -l(y) = = ( B ( h 1 ) f and | f y ( h . ) | - 1 - ? i ( y ) p - ^ 0 . | y W h l | . 128 d) If y, z € ( H) a n d y * z, there e x i s t s j € J such that (y, h j ) g ^ ( z/ n j ^ H a n d ' h e n c e < h j ' y > H ^ < h j ' z > -Thus, .y H(y) * y H ( z ) . a), b ) , c) show that ^ i s an i s o m e t r i c , and t h e r e f o r e con-t i n u o u s , operator from (H) with i t s *norm to X*. d) shows that i s one-to-one. The l a t t e r part of the theorem f o l l o w s from the l i n e a r i t y of ^pH and Theorem 2 . 3 ( i i i ) . Def i n i t i o n 3 - 3 * A subset A of a V—space i s c a l l e d l o c a l l y f i n i t e i f f o r every i n t e g e r n, there i s at most a f i n i t e number of elements of A with norms equal to p U . Lemma 3.U- • Let f £ x * ' If the subset H' of H on which f i s non-zero i s bounded and l o c a l l y f i n i t e , then ( i ) H f i s a f i n i t e set; ( i i ) there e x i s t s y f C (H) such that ^ H ( y f ) E f-Proof : ( i ) Since f i s a continuous l i n e a r mapping i n t o the d i s c r e t e space F, there e x i s t s an i n t e g e r m such that f ( x ) £ 0 i m p l i e s | x | > p™. Thus, H T i s bounded below, bounded above and l o c a l l y f i n i t e ; hence i t ' i s f i n i t e . ( i i ) Let (6.6) y f = £ f(h»)h». h»€H» For a l l h ^ H / 129 f (h) = <h, y f > H = <p H ( y f ) ( h ) . From Theorem 4 -4«l, i t f o l l o w s that ^>jj(yf) = f» Theorem 3 »5« The operator ^ i s an i s o m e t r i c isomorphism be-tween (H) with i t s *norm and X* i f and only i f X i s bounded and H i s l o c a l l y f i n i t e . P r oof: If X i s bounded and H i s l o c a l l y f i n i t e , every subset H* of H s a t i s f i e s the hypotheses of Lemma 3«4« Therefore, ipjj maps (H) onto X*. For the converse, suppose that X i s unbounded or that H i s not l o c a l l y f i n i t e . Then, f o r some i n t e g e r n there e x i s t s an i n f i n i t e subset H T of H such that f o r a l l h' £ H': I*1* 1 — p°'t when X i s unbounded, or |h» | = p , when H i s not l o c a l l y f i n i t e . From Theorem 4-4=1, there e x i s t s f £ X* such that f ( h ' ) = 1 f o r a l l h» £ H», f ( h ) = 0 f o r a l l h £ H^H" Should there e x i s t y^ g (H) such that ^ ( y ^ ) s f, y^ would have to have the i n f i n i t e expansion (6.6). Th i s i s i m p o s s i b l e . C o r o l l a r y 3°6° If X i s unbounded and H i s l o c a l l y f i n i t e , then jpjj i s an i s o m e t r i c isomorphism between (H) with i t s *norm and the subspace of X* formed by the continuous l i n e a r f u n c t i o n a l s which vanish outside of a bounded subset of X. C o r o l l a r y 3.7. If X d i s t i n g u i s h e d b a s i s , i s bounded then X and and admits a l o c a l l y f i n i t e X* have the same dimension. 1 3 0 The proof f o l l o w s from Theorems 3 - 2 and 3 •> 5 <> Examples : The spaces ( ? k of 3 - 4 and Q k of 3 - 5 are bounded and admit l o c a l l y f i n i t e d i s t i n g u i s h e d bases. Thus, the spaces ^ £ and Q£ are e q u i v a l e n t to the spaces of polynomials ( 6 . 7 ) and ( 6 . 8 ) r e s p e c t i v e l y : ( 6 . 7 ) | (A.) s 0 i=p I? I* = 0 -* = p n , a n * 0 , k < p < n. (6.8) ' f (u, v) = 0 * = 0 , i=p j= 0 ~ f ' I |a h I * 0 , k < p < n. j= 0 n J A c c o r d i n g to ( 6 . 5 ) , a continuous l i n e a r f u n c t i o n a l on (P^ i s a f i n i t e l i n e a r combination of the f u n c t i o n a l s f : f (x) = c o e f f i c i e n t of A,n i n the expansion of x(A,) ?n in powers of A,. This r e s u l t was proved d i r e c t l y by H. F„ Davis [ 4 ] , P» 9 1 , f o r the space CP It was shown i n 3 - 4 that CP^ admits as d i s t i n g u i s h e d bases the sets § Q and J of ( 3 . 1 6 ) , Ch. 3 . Consider the continuous l i n e a r f u n c t i o n a l f d e f i n e d on ^ q by 131 f(<p n) = ( a n * 0 f o r 0 < n < N, „ 0 f o r n > N. The i s o m e t r i c isomorphism -J-^ between ) and @ * i s such that • o N t i 0 ( f ) • I " " V 0 n=0 The i s o m e t r i c isomorphism -IrT between (J) and CP* i s such that N n=0 where the c o e f f i c i e n t s B^, determined from the power s e r i e s expansions of the J R ' s ( [ 8 ] ) , are the s o l u t i o n s of the system: p - i 2 i i * 0 22 p ( p - i ) ! ( p + i ) l 2 p 2 V ( - l ) p _ i " 2 i + 1 = a ' it-Q 2 2 p + 1 ( p - i ) ! ( p + i n ) ! 2 p + 1 ' P = o, i , C l e a r l y , i n "frfiS^^) ^ ^ j ^ ^ 1 ^ 0 This i n e q u a l i t y r e f l e c t s the dependence of the H-inner product on the d i s t i n g u i s h e d b a s i s H. 0 1 3 2 B i b l i ography [ l ] M. G. Arsove, The Paley-Wiener Theorem i n m e t r i c l i n e a r  spaces, P a c i f i c J . Math., 10 (I960) pp. 365-379. [ 2 ] C. W. Clenshaw, The numerical s o l u t i o n of l i n e a r d i f f e r e n -t i a l equations i n Chebyshev s e r i e s , Proc. Camb. P h i l . S o c , 5 3 ( 1 9 5 7 ) P P . 1 3 4 - 1 4 9 = [ 3 ] !• S. Cohen, On non-Archimedean normed spaces, Proc. Kon. Ned, Akad. Wet., 5 1 ( 1 9 4 8 ) pp. 6 9 3 - 6 9 8 . [ 4 ] H..F. Davis, A note on asymptotic s e r i e s . Can. J . Math., 9 ( 1 9 5 7 ) P P . 9 0 - 9 5 . ' [ 5 ] J . de Groot, Non-Archimedean metrics i n topology, Proc. Amer. Math. S o c , 7 ( 1 9 5 6 ) pp. 9 4 8 - 9 5 3 -. [ 6 ] V. A. D i t k i n and A. P. Prudnikov, "Operati onal C a l c u l u s i n two v a r i a b l e s and i t s a p p l i c a t i o n s " , ( E n g l i sh t r a n s l a -t i o n by D. M.' G. Wi s h a r t ) , Pergamon Press, New York, 1 9 6 2 . [ 7 ] N. Dunford and J . T. Schwartz, " L i n e a r Operators, Part I; General Theory", I n t e r s c i e n c e P u b l i s h e r s Inc., New York, 1 9 5 8 . [8] A. E r d e l y i et a l . , "Tables of I n t e g r a l Transforms", V o l . I and V o l . I I , McGraw-Hill Book Co. Inc., New York, 1 9 5 4 , [ 9 ] A. E r d e l y i , "Asymptotic Expansions", Dover P u b l i c a t i o n s , Inc., New York, 1 9 5 6 . [ 1 0 ] A. E r d e l y i , General asymptotic expansions of Laplace i n t e g r a l s . Arch. Rat. Mech. A n a l . . 7 ( 1 9 6 1 ) pp. 1 - 2 0 . [ 1 1 ] A. E r d e l y i and M. Wyman, The asymptotic e v a l u a t i o n of cer- t a i n i n t e g r a l s , Arch"!! Rat. Mech. Anal., 14 ( 1 9 6 3 ) pp. 2 17 - 2 6 0 . [ 1 2 ] I. F l e i s c h e r , Sur l e s espaces normes non-archimediens, Proc. Kon. Ned. Akad. Wet, Ser. A, 5 7 ( 1 9 5 6 ) pp. 1 6 5 -168. [ 1 3 ] L. Fox, Chebyshev '-methods f o r ordinary d i f f e r e n t i a l equa- t i o n s , Computer Jnl.', 4 ( 1 9 6 1 - 6 2 ) pp. 318-331. [ 1 4 ] P. R. Halmos, " I n t r o d u c t i o n to H i l b e r t spaces", 2 n d ed., Chelsea Publ. Co., New York, 1 9 5 7 . 133 [ 1 5 ] [16] [ 1 7 ] [18] [ 1 9 ] [ 2 0 ] [ 2 1 ] [ 2 2 ] [ 2 3 ] [ 2 4 ] [ 2 5 ] T. E. Hu 11, On some i n t e g r a l equations with unbounded eigenv a l u e s , S.I.A.M. J o u r n a l , 7 ( 1 9 5 9 ) pp. 2 9 0 - 2 9 7 . W. Hurewicz and'H. Wallman^ "Dimension Theory", P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , N, J . , 1 9 4 1 . A. W. I n g l e t o n , The Hahn-Banach Theorem f o r non-Archimedean valued f i e l d s , Proc. Camb. P h i l o . Soc. 4 8 ( 1 9 5 2 ) pp. 4 1 -J . L. K e l l e y , "General topology", D. Van Nostrand Co. Inc., P r i n c e t o n , N. J . , 1 9 5 5 . A. N. Kolmogorof and S. V. Fomin, "Elements of the Theory of Functions and F u n c t i o n a l A n a l y s i s " , V o l . I and V o l . II (Russian) E n g l i s h t ran s l a t i on, Gray lock Press, Rochester, N. Y., 1 9 5 7 . C. Lanczos, Trigo n o m e t r i c i n t e r p o l a t i o n of e m p i r i c a l and  a n a l y t i c a l f u n c t i o n s , J . Math, and Phys., 17 ( 1 9 3 8 ) pp. 1 2 3 - 1 9 9 . C. Lanczos, " A p p l i e d A n a l y s i s " , P r e n t i c e H a l l Inc., Englewood C l i f f s , N. J . , 1 9 5 6 . L. H. Loomis, "An I n t r o d u c t i o n to A b s t r a c t Harmonic Analysis"', D. Van Nostrand, Inc., New York, 1 9 5 3 . J . M i k u s i n s k i , " O p e r a t i o n a l C a l c u l u s " , ( T r a n s l a t i o n of the 2 n d P o l i s h e d i t i o n , v o l . 3 0 , Monografie Matematyczne, 1 9 5 7 > , Pergamon Press, Inc., New York, 1 9 5 9 . A. F. Monna, Sur l e s espaces l i n e a i r e s normes. I : Proc . Kon . Ned. Akad. Wet., 4 9 ( 1 9 4 6 ) pp. 1 0 4 5 - 1 0 5 5 ; II : t» i t i t tt i t 4 9 ( 1 9 4 6 ) pp • 1056-1062; I I I : n t t t t t t t t 4 9 ( 1 9 4 6 ) pp. 1 1 3 4 - H 4 1 ; IV: It t t tt t t i t 4 9 ( 1 9 4 6 ) pp. 1 1 4 2 - 1 1 5 2 ; V : »/ n t t i t i t 5 1 ( 1 9 4 8 ) pp. 1 9 7 - 2 1 0 ; VI : t t t t t t tt, t i 52 ( 1 9 4 9 ) pp. 151- 160. A. F. Monna, Sur l e s espaces normes non-archimediens, I: Proc. Kon. Ned. Akad. Wet, Ser. A, 5 9 ( 1 9 5 6 ) pp. 4 7 5 - 4 8 3 ; I I : " "• "• '* " " " 5 9 ( 1 9 5 6 ) I I I : IV: t t 11 t t t t 11 t t 11 tt i t tt i t i t pp. 4 8 4 60 ( 1 9 5 7 ) PP. 4 5 9 60 ( 1 9 5 7 ) pp. 4 6 8 4 6 7 ; 4 7 6 . 4 8 9 ; I I 1 3 4 [26] M. A. Nairaark, "Normed r i n g s " , ( E n g l i s h t r a n s l a t i o n of the f i r s t Russian ed. by L. F. Boron), P. Noordhoff, N. V., Groningen, Netherland, 1 9 5 9 » [27] R. E. A. C. Paley and N. Wiener, " F o u r i e r Transforms i n the Complex Domain", Amer. Math. Soc. Colloquium Publ. no. 1 9 , New York, 1 9 3 4 -[28] L. P o n t r j a g i n , " T o p o l o g i c a l Groups", E n g l i s h t r a n s l a t i o n , P r i n c e t o n Univ. Press, P r i n c e t o n , N. J . , 1 9 4 6 . [29] Popken, Asymptotic expansions from an a l g e b r a i c stand-p o i n t , Proc. Kon. Ned. Akad. Wet, Ser. A, 56 (1953) pp. I 3 I - I 4 3 . [ 3 0 ] F. W. Schafke, Das K r i t erium von Paley und Wiener in  Banaschem Raum, Math. Nach., 3 ( 1 9 4 9 ) pp. 59-61. [ 3 1 ] 0 . F. G. S c h i l l i n g , "The Theory of V a l u a t i o n s " , Mathematical Surveys, No. IV, Amer. Math. S o c , 1 9 5 0 . [ 3 2 ] M. S c h u l z e r , "Asymptotic P r o p e r t i e s of S o l u t i o n s of Equa- t i o n s i n Banach Spaces", M.A. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 1 9 5 9 . [33] C. A. Swan son and M.. S c h u l z e r , Asymptotic s o l u t i o n s of equations i n Banach spaces, Can. J . of Math., 13 (1961) pp. 4 9 3 - 5 0 4 . [34] C. A. Swanson, An i n e q u a l i t y f o r l i n e a r t r a n s f o r m a t i o n s  with e i g e n v a l u e s . B u l l . Ame r. Math . S o c , 67 ( 1 9 6 1 ) pp. 607-608. [35] C. A. Swanson, On s p e c t r a l e s t i m a t i o n . B u l l . Amer. Math. S o c , 6 8 ( 1 9 6 2 ) pp. 33-35 . [ 3 6 ] A. E. T a y l o r , " I n t r o d u c t i o n to F u n c t i o n a l A n a l y s i s " , J . Wiley and Sons, Inc., New York, 1 9 5 8 . [37] J . Todd, " I n t r o d u c t i o n t o the C o n s t r u c t i v e Theory of Func- t i on s". I n t e r n a t i o n a l S e r i e s of Numerical Mathematics, V o l . I, Academic Press Inc., New York, I 9 6 3 . [ 3 8 ] J . G. van der Corput, "Asymptotic Expansions I. Funda- mental Theorems of Asymptotics", U n i v e r s i t y of C a l i f o r n i a , B erkeley, 1 9 5 4 . [39] J . G. van der Corput, Asymptotics. I, Proc. Kon. Ned. Akad. Wet, Ser. A, 5 7 ( 1 9 5 4 ) PP• 2 0 6 - 2 1 7 . 135 [ 4 0 ] B. L. van der Waerden, "Modern Alg e b r a " , V o l . I, Revised E n g l i s h e d i t i o n , F r e d e r i c k Ungar Publ. Co., New York, 1953-[ 4 1 ] D. V. Widder, "The Laplace Transform". P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , N. J . , I 9 4 I . 1 136 Notation Index F 6 * 61 0 6 62 S ( x , r ) , S ' ( x , r ) , B(x, r) 9 * • * > ^ k 63 H ( A ) 17 Q 65 (A), [A] 22 * k 66 (x, y ) A 28 67 W(A) 30 71 P 3 0 , 31 I AJ Z, f A | 78 X © Y 39 er ( z , Y ) , €T(x) 79, A d 40 3(z, Y ) , ^ ( X ) 79, 45 0, I 80 e 49 X * 121 O-(x) 52 J(x) 124 A, s, A,Q 52 oo(x), l ( x ) 124 cd-nbhd 53 M * 124 f = 0 ( g ) , f = o(g) 54 J(x, y) 126 B, B Q, N 54 <x, y> H 126 i 55 ?K 127 0- J(n) 55 y f 128 1 3 7 Subj ect Index Asympt ot i c norm, 5 5 s c a l e , 5 5 sequence, 5 4 space, 5 4 -A s y m p t o t i c a l l y f i n i t e , 55 Bases complete, 2 3 d i s t i n g u i s h e d , 25 e x i s t e n c e of d i s t i n g u i s h e d , 3 2 extension of, 4 1 Hamel, 2 3 Cauchy sequence, s e r i e s , 16 Cd-nbhd, 5 2 C h a r a c t e r i z a t i o n of l i n e a r o p e r a t o r s , 8 8 -Compactness, 1 7 -Complete s p e c t r a l decomposition d e f i n i t i o n of, 9 5 comparison of, 9 8 and dimension of the space, 9 7 Completely independent set, 2 3 Completeness of asymptotic spaces, 5 6 Dimension of a V-space, 2 9 D i r e c t sum of subspaces, 3 9 ' D i s t i n g u i shed adjunct, 4 0 b a s i s , 2 5 complements, 4 1 e x i s t e n c e of - bases, 3 2 f a m i l y of s e t s , 3 5 -set s, 24 Dual space X*, 1 2 1 boundedness and d i s c r e t e n e s s of, 1 2 3 i s o m e t r i c isomorphism of - with (H), 1 2 7 , 1 2 9 E q u a t i o n s : see " S o l u t i o n s of -" Expansion i n terms of, 28 Expansion of the Poincare type, 60 E x t e n s i o n of a b a s i s , 4 1 F o u r i e r c o e f f i c i e n t s , s e r i e s , 7 4 F u n c t i o n a l , 1 2 1 -138 Hahn-Banach theorem, 1 2 1 , 1 2 2 H-Lnner product, 126-(H) •norm on, 1 2 3 i s o m e t r i c isomorphism of - with X*, 1 2 7 , 1 2 9 I d e n t i t y i n a V-algebra, 4 5 pseudo- , - i n a V-algebra, 4 6 operator, 80 pseudo-, - i n U ( X ) , 9 4 Inner product: see "H-inner p r o d u c t " Inverse i n a V-algebra, 4 7 pseudo-, - i n a V-algebra, 4 6 (pseudo-) - of an operator on i t s range, 8 4 (pseudo-) - of an operator i n 3 (X), 9 4 see " S o l u t i o n s of e q u a t i o n s " Isometric isomorphism d e f i n i t i o n , 1 2 6 between ( H ) and X*, 1 2 7 , 129 L i n e a r f u n c t i o n a l s , 1 2 1 r e p r e s e n t a t i o n of continuous, 127, 1 2 9 L i n e a r l y n o n - t r i v i a l set, 8 3 L o c a l l y f i n i t e s e t , 128 Moment of a f u n c t i o n , 7 1 space, 7 2 examples of - spaces, 7 5 N on-Arch imedean m e t r i c , 7 s t r o n g l y - m e t r i c , 7 N orm of an operator, 7 8 p r o p e r t i e s of, 1 -range, 17 *norm, 1 2 3 N o t a t i o n a l conventions on p, 3 1 on = and s, 4 5 0 r e l a t i o n s , 5 2 o r e l a t i o n s , 5 2 Operators bounded, 7 8 boundedness and c o n t i n u i t y of, 8 5 , 8 6 c h a r a c t e r i z a t i o n of l i n e a r , 8 8 1 3 9 d e f i n i t i o n of l i n e a r , 7 8 i n v e r s e s : see " I n v e r s e " spectrum of - i n If (X), 9 4 see "Complete s p e c t r a l decomposition Paley-Wiener theorem, 3 4 Parameters, primary and secondary, 5 2 , 5 3 P o i n c a r e : see "Expansion" P r o j e c t i o n s , 1 0 2 -Regular element of a V-algebra, 4 6 pseudo-, 4 6 Riesz's Lemma, 2 0 S i n g u l a r element of a V-algebra, 4 7 S o l u t i o n s of equations, 1 0 5 -xz ' = y i n V-algebras, 1 0 5 -xz = e i n V-algebras, 1 0 5 -Ax = y i n V-spaces, 1 1 1 - , 1 1 5 -d e f i n i t i o n of pseudo--, 1 1 1 Spectrum, 4 9 S p h e r i c a l completeness, 4 4 Subspace ' generated by A, (A), 2 2 c l o s e d - generated by A, [ A ] , 2 2 •norm, 1 2 3 -T r i v i a l s e t , 3 5 -T r i v i a l v a l u a t i o n , 7 s Uniform Boundedness P r i n c i p l e , 8 6 , 8 7 Usual topology on R and C, 1 2 Usual v a l u a t i o n on R and C, 12 V-algebra, d e f i n i t i o n of, 4 5 V-space d e f i n i t i o n , 30 d i s c r e t e , 2>0 of bounded operators, 0"(Z, Y), 6f(X), 80-of bounded l i n e a r o p e r a t o r s , ^ (Z, Y), 3(X), 8 : 3 -Valued space def i n i t i on of , 6 pseudo- -, 6 s t r o n g l y (pseudo-)-, 6 Zero-dimensional space, 1 1 

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