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An extension of the Krein-Milman theorem and applications Kirshner, David 1977

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AN EXTENSION OF THE KREIN-MILMAN THEOREM AND APPLICATIONS by DAVID KIRSHNER B.A., S i r George W i l l i a m s U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE: STUDIES 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 BRITISH COLUMBIA September, 1977 MASTER OF ARTS i n D a v i d K i r s h n e r , 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree 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 r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f 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 a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Department o f ^ i - h s m a t i e s The U n i v e r s i t y o f B r i t i s h Co lumbia 20 75 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 flfifehftf A, 1977 i i ABSTRACT The K r e i n - M i l m a n Theorem s a y s t h a t each compact, convex s u b s e t o f a l o c a l l y convex space i s t h e c l o s e d convex h u l l o f i t s extreme p o i n t s . I n t h e c a s e of a s e p a r a b l e Banach Space s e v e r a l c o l l e c t i o n s o f extreme p o i n t s a r e known t o be dense i n t h e whole s e t o f extreme p o i n t s ( e . g . t h e s e t o f exposed p o i n t s [5; theorem 4]; t h e s e t o f d e n t i n g p o i n t s [8; remarks f o l l o w i n g definition44]). C o n s e q u e n t l y t h e s e s e t s can be used i n s t e a d o f t h e whole s e t of extreme p o i n t s t o g e n e r a t e compact convex s e t s . I n t h i s t h e s i s we examine such a dense s u b s e t o f extreme p o i n t s i n t h e c o n t e x t o f l e s s s t r u c t u r e d s e p a r a b l e l o c a l l y convex s p a c e s . We a l s o examine some a p p l i c a t i o n s o f t h e r e s u l t i n g e x t e n d e d K r e i n - M i l m a n Theorem. i i i TABLE OF CONTENTS C h a p t e r Page Name 0 1 P r e l i m i n a r y N o t a t i o n s , D e f i n i t i o n s , and Theorems 1 16 The C h i p p i n g Lemma and t h e K r e i n - M i l m a n E x t e n s i o n Theorem 2 55 A p p l i c a t i o n s B i b l i o g r a p h y i v ACKNOWLEDGEMENTS I am g r e a t f u l t o my s u p e r v i s o r Dr. E. E. G r a n i r e r f o r s u g g e s t i n g t h e t o p i c t o me, and f o r h i s p a t i e n c e , encouragement and f r i e n d s h i p d u r i n g t h e w r i t i n g o f t h e t h e s i s . I am indehtfed a l s o t o N e i l F a l k n e r f o r many h e l p f u l c o n v e r s a t i o n s c o n c e r n i n g t h e s u b j e c t o f t h i s p a p e r . F i n a l l y I w i s h t o thank M i s s I v y Thomas f o r h e r work i n t y p i n g the t h e s i s . INTRODUCTION 1 T h i s p a p e r m i g h t w e l l have been c a l l e d t h e c h i p p i n g Lemma and i t s a p p l i c a t i o n s r a t h e r t h a n i t s p r e s e n t t i t l e , s i n c e t h e c h i p p i n g lemma i s th e main t o o l i n e x t e n d i n g t h e K r e i n - M i l m a n theorem. B r e i f l y , g i v e n a s e p a r a b l e , H a u s d o r f f , l o c a l l y convex space and a c o n t i n u o u s pseudonorm p d e f i n e d on i t , t h e c h i p p i n g lemma s t a t e s t h a t f o r any compact, convex s e t , a s m a l l ( o f p - d i a m e t e r l e s s t h a n a g i v e n e > 0) b u t non-empty s e t can be " c h i p p e d away" l e a v i n g i n t a c t t h e p r o p e r t i e s o f compactness and c o n v e c i t y o f t h e r e m a i n d e r . I n c h a p t e r 1 o f t h i s p a p e r t h i s r e s u l t o f I.Namioka and E. A s p l u n d [2] i s m o t i v a t e d and s l i g h t l y g e n e r a l i z e d . The p r o o f i n [2] of t h e R y l l - N a r d z e w s k i f i x e d p o i n t theorem u s i n g t h e c h i p p i n g lemma i s r e p r o -duced, and some a p p l i c a t i o n s o f i t a r e examined t o m o t i v a t e n t h i s l i n e o f r e s e a r c h h i s t o r i c a l l y . F o l l o w i n g Namioka i n [ 1 ] , t h e lemma i s t h e n a p p l i e d t o e x t e n d the K r e i n - M i l m a n theorem. The method o f p r o o f h e r e i s t h e same as Namioka's, however t h e n o t a t i o n has been somewhat s i m p l i -f i e d , and t h e p r o c e d u r e s m o t i v a t e d . We b e g i n c h a p t e r 2 o f a p p l i c a t i o n s by e x a m i n i n g s p e c i f i c l o c a l l y convex h a u s d o r f f spaces t o w h i c h t h e ex t e n d e d K r e i n - M i l m a n theorem can be n o n - t r i v i a l l y a p p l i e d , and as i n [ 1 ; theorem 2.3] t h e r e s u l t i s r e -f o r m u l a t e d i n g r e a t e r g e n e r a l i t y . N e x t , d i v e r s e a p p l i c a t i o n s a r e ex-amined. F o r example t h e K r e i n - M i l m a n theorem i s e x t e n d e d i n a d i f f e r e n t d i r e c t i o n [ 1 ; theorem 3.6] — i n a F r e c h e t space where second d u a l i s q u a s e - s e p a r a b l e r e l a t i v e t o t h e s t r o n g t o p o l o g y , the c l o s e d , bounded (n o t n e c e s s a r i l y c o m pact), convex s u b s e t s a r e t h e c l o s e d convex h u l l o f t h e i r extreme p o i n t s . T h i s and o t h e r a p p l i c a t i o n s f r o m [ 1 ; s e c t i o n 3] a r e s i m p l i f i e d b y the a d d i t i o n o f numerous d e t a i l s t o t h e p r o o f s . We 2 c o n c l u d e w i t h a s l i g h t g e n e r a l i z a t i o n o f t h e R y l l - N a r d z e u r k i f i x e d p o i n t theorem [ 1 ; theorem 3 . 7 ] . C h a p t e r 0, t h e i n t r o d u c t i o n , p r o v i d e s ample b a c k g r o u n d t h e o r y t h r o u g h w h i c h t h i s t h e s i s s h o u l d be a c c e s s i b l e t o any s t u d e n t h a v i n g a f i r s t f u n c t i o n a l a n a l y s i s c o u r s e . 3 Section 0 Preliminary Notations Definitions and Theorems Notations: Number Spaces IN'v • denotes the set of natural numbers, fft denotes the set of feal numbers. C denotes the set of complex numbers. C n denotes the Cartesian product of C with i t s e l f n-times. inf(X) i s the infemum of a set X of real numbers. (a'»B) 9 [ a>3] denotes-the open, closed intervals from a to 3 , respectively. t TSet.Theory AXB i s the Cartesian product of the sets A and B IT P. is Cartesian product of the sets p. i £ I A -- B i s the set of elements which are in the set A but not in the set B comp(A) is the complement of the set A Topo'lTppology C1(K) denotes the closure of the set K int(K) denotes the interior of the set K G. -sets are sets formed by taking countable intersections of open o sets. 4 L i n e a r A l g e b r a dim (X) denotes t h e d i m e n s i o n o f v e c t o r space X C 0 ( K ) " .-.//denotes the s e t o f convex c o m b i n a t i o n s o f a s u b s e t K o f some v e c t o r s p a c e , r a t i o n a l convex c o m b i n a t i o n s r e f e r s t o convex c o m b i n a t i o n s w i t h r a t i o n a l c o e f f i c i e n t s . F u n c t i o n a l A n a l y s i s F o r ( X , J ) a t o p o l o g i c a l v e c t o r space t h e s c a l a r f i e l d o f X w i l l be presumed t o be E. u n l e s s o t h e r w i s e s t a t e d . By a w e l l known r e s u l t [6; remarks p r e c e d i n g theorem 3 . 2 ] , t h e t h e o r y h e r e i n p r e s e n t e d a p p l i e s t o complex t o p o l o g i c a l v e c t o r s paces as w e l l . A s u b s e t K o f X w i l l be d e s c r i b e d as J - c l o s e d (open e t c . ) t o i n d i c a t e d t h a t K i s c l o s e d (open e t c . ) i n (X,J) . S i m i l a r l y a c o n -t i n u o u s ( l o w e r s e m i c o n t i n u o u s , e t c . ) f u n c t i o n on ( X , J ) w i l l be des-c r i b e d as J - c o n t i n u o u s ( l o w e r J - s e m i c o n t i n u o u s , e t c . ) on X Oth e r T o p o l o g i e s on X F o r ( X , J ) a l o c a l l y convex t o p o l o g i c a l v e c t o r space ( X , J ) * ( o r X* when J i s u n d e r s t o o d ) w i l l denote t h e c o n t i n u o u s d u a l o f ( X , J ) The weak t o p o l o g y on (X,J) w w i l l be denoted by ( X , J W ) T o p o l o g i e s an. X* T o p o l o g i e s on X* ( X , J ) * w i t h t h e w e a k - s t a r t o p o l o g y w i l l be denoted (X*,w*) ( X , J ) * w i t h t h e s t r o n g t o p o l o g y i s denoted as (X*,s) and i s t h e t o p -o l o g y o f u n i f o r m convergence on bounded s e t s i n (X,J) . That i s : a 5 l o c a l b ase f o r (X*,s) i s g i v e n by {B° : B i s bounded i n X B° = { f £ X* : f (x) _< 1 f o r e v e r y x B } i s t h e p o l a r o f O t h e r D e r i v e d T o p o l o g i e s F o r (X,Ji) a t o p o l o g i c a l v e c t o r space and K C X , (K,J) K w i t h t h e i n d u c e d t o p o l o g y . DEFINITIONS Pseudo Nosnms 0.1 D e f i n i t i o n : A pseudo-norm ( c a l l e d semi norms by some a u t h o r s ) p on a v e c t o r space X i s a map, p: X —>- [0,°°) s u c h t h a t f p ( x + y) i p ( x ) + p ( y ) \ f o r each x,y e X and\p(ax) = | a| p ( x ) J - and - aa £ (R I I 0.2 D e f i n i t i o n : L e t (X,J) be a t o p o l o g i c a l v e c t o r space and p a pseudo-norm on X . P i s l o w e r J - s e m i c o n t i n u o u s i f and o n l y i f {x: p ( x ) ^-1} i s J - c l o s e d . 0.3 Remark: A pseudognoiEmffi, p on a v e c t o r space X g e n e r a t e s a t o p -o l o g y on X i n t h e same.manner as does a norm. That i s , a l o c a l base f o r t h e t o p o l o g y i s g i v e n by { B e } E > 6 where B £ * { x : p ( x ) < e } Of c o u r s e t h i s t o p o l o g y i s n o t i n g e n e r a l h a u s d o r f f . 0.4 N o t a t i o n : The t o p o l o g y d e r i v e d f r o m pseudo-norm p w i l l be d e n oted Jp . 0.5 D e f i n i t i o n : L e t ^p } a £ j be a s e t °^ pseudo-norms on a l o c a l l y c o n -v e x t o p o l o g i c a l v e c t o r space (X , J ) . ^ P a ^ a e x d e t e r m i n e s J means p. I } where B . d e n o t e s t h a t a n e t {xg} w i l l J - c o n v e r g e t o some x i n X p r e c i s e l y when l i m p (xg - x) = 0 f o r e v e r y a e I <$• a 0.6 D e f i n i t i o n : L e t A be a convex, b a l a n c e d a b s o r b i n g s e t i n a v e c -t o r s pace X . The Minkowsky f u n c t i o n a l on / A - (denoted u ) i s de-f i n e d by ^ A ( x ) = i n f { t > 0: t _ 1 x e A } . (By [6: theorem 1.35] i t i s e a s i l y seen t h a t U i s a pseudo-norm on X . 0.7 D e f i n i t i i i o n : A pseudo m e t r i c Cd on a v e c t o r space X l i s a map dw XxX —*• [0,°°) such t h a t : . - d ( x , x ) = 0 •id(:x.yy) ='d'{y,x) andndCd(x,y) £Cd7.((sc,z) + d ( z , y ) f o r each x,y,z G X Cd ( x , y ) = 0 whe're xx-*^yy mmayooccur. g e n e r a t e s a t o p o l o g y on X i n t h e u s u a l way. P r o p e r t i e s o f S e t s 0.8 D e f i n i t i o n : A s u b s e t o f a t o p o l o g i c a l space i s s a i d t o be nowhere  dense i f i t s c l o s u r e has empty i n t e r i o r . 0.9 D e f i n i t i o n : A s u b s e t o f a t o p o l o g i c a l space S i s o f f i r s t c a t e - g o r y i n S i f and o n l y i f i t i s a c o u n t a b l e u n i o n o f nowhere dense s e t s 0.10 D e f i n i t i o n : A s u b s e t o f a t o p o l o g i c a l space S i s s a i d t o be o f second c a t o g o r y i n S p r o v i d e d t h a t i t i s n o t o f f i r s t c a t e g o r y i n S PRELIMINARY THEOREMS C l a s s i c a l R e s u l t s The f i r s t f o u r w e l l known theorems a r e s t a t e d ( w i t h o u t p r o o f ) due to their importance in the results of this thesis. 0.11 Theorem: (Krein-Milman) [6; theorem 3.21] Suppose X is a locally convex topological vector space. If K £ X is compact and convex, then K i s the closed convex h u l l of i t s extreme points. 0.12 Theorem: [13; V.8.3 Lemma 5 Page 440] Let Q be a compact set in a locally convex linear topological space X whose closed convex h u l l i s compact. Then the only extreme points of C£[Cg(Q)] are points of Q 0.13 Theorem: (Baire) [6; theorem 2.2] If S is either (a) a complete metric space, or (b) a locally compact Hausdorff space, then the intersection of every countable collection of dense open sub-sets of S i s dense in S 0.14 Theorem: (Markov-Katetani).) [13: V.10.5 Page 456 theorem 6] Let R be a compact convex subset of a Linear topological space X . Let F be a commuting family of continuous linear mappings which map K into i t s e l f . Thenkthere exists a point p <£. K such that Tp = p for each T e F . CATEGORY THEOREMS 0.15 Definition: Any space which satisfies the conclusion of Baire's Theorem is called a Baire Space. 0.16 Remark; E v e r y B a i r e space S i s o f second c a t e g o r y i n i t s e l f . 00 I n d e e d , t h e a s s u m p t i o n t h a t S =.U,C_- where each C. i s nowhere dense i = l 1 l 00 means t h a t s = Cj.jCtCC^ where int[C£(C ) ] = 0 f o r e v e r y C± . C l e a r l y comp(C£(C i>) i s open dense i n S and thus ^ comp (Ct(C/)) CO i s dense i n S , hence non-empty. C o n s e q u e n t l y ^U^ CZ(C/) / S and t h e a s s u m p t i o n i s c o n t r a d i c t e d . To d e a l w i t h l o w e r J - s e m i c o n t i n u o u s pseudo-norms on a t o p o l o g i c a l v e c t o r space ( X , J ) , and t h e subsequent J - c l o s e d s u b s e t s we r e s t a t e p a r t o f B a i r e ' s theorem f o r c l o s e d s e t s . 0.17 C o r o l l a r y ; [ 1 ; lemma 1.1] L e t X be a compact H a u s d o r f f s p a c e . L e t {C.}. be a c o u n t a b l e c o l l e c t i o n o f c l o s e d s u b s e t s o f X such t h a t X =.u,C. . Then . U , i n t ( C . ) i = l l i = l - i y is i.densg" MnsX--i,g. -X . P r o o f : L e t U ^ 0 be an a r b i t r e r y open s e t i n X . Then (U,J) 00 i s a l o c a l l y compact H a u s d o r f f space and U =.U,U n C. where each U OC. i = l l l i s c l o s e d i n (U,J) . But (U,J) i s a B a i r e space and hence second c a t e g o r y i n i t s e l f (remark 0.15) t h u s D f l C , has non-empty i n t e r i o r 00 f o r some C. and s i n c e U was a r b i t r a r i l y c h o s e n , .U..C. i s dense i n i J i = l l (X,J) . EVALUATION MAPS 0.18 D e f i n i t i o n : L e t E be a t o p o l o g i c a l v e c t o r s p a c e , and E* i t s d u a l . F o r e v e r y x e X d e f i n e F : E* —> IR by F ( f ) = f ( x ) . Then X X F £ ( E * , s ) * and the map I : E - > ( E * , s ) * d e f i n e d by I ( x ) = F c a l l e d x x t h e e v a l u a t i o n map on E i s one-to-one. 9 The f o l l o w i n g theorem and i t s p r o o f a r e adopted f r o m [10; 33.2 page 346] and [10; 36.5 page 3 7 3 ] . 0.19 Theorem: F o r E a F r e c h e t Space, t h e e v a l u a t i o n map T: 1 (E*-fs [ (fE*} s) s<]:on±sncontinuous. P r o o f : L e t V be a 0-neighbourhood i n [ ( E * , s ) * , s ] . S i n c e {B°: B i s s t r o n g l y bounded i n E*} forms a l o c a l b ase f o r [ ( E * , s ) * , s ] we can f i n d B° £ V , B s t r o n g l y open i n E* . I'^CB 0) = {x: | f ( x ) | •<_ 1 f o r e v e r y f e B} i s a b s o r b i n g i n E . Indeed i f y £ E t h e n by t h e c o n t i n u i t y o f s c a l a r m u l t i p l i c a t i o n ' {y} i s bounded i n E and so {y}° i s a s t r o n g 0-neighbourhood i n E* S l i n c e B i s s t r o n g l y bounded i n E* t h e r e e x i s t s m > 0 such t h a t B £ m { y } ° = m { f : | f ( y ) | < 1} = { f : | f ( y ) | < m} . Thus y e m f ^ B 0 ) . y b e i n g a r b i t r a r i l y c h osen, we g e t t h a t E = ^ ml "*"(B0) and ml "''(B0) i s c l o s e d i n E f o r e v e r y m . But E i s F r e c h e t , hence a B a i r e Space, hence second c a t e g o r y i n i t s e l f , (0.15 r e m a r k ) . Thus ml "SjB°) has non-empty i n t e r i o r f o r some m , and so I "*"(B°) has non-empty i n t e r i o r . But 0 i s i n t e r n a l t o I "*"(B°) i n the a l g e b r a i c s ense ( i . e . each l i n e p a s s i n g t h r o u g h 0 has some s e g -ment i n I ''"(B 0)) fianarhenceaby.- a w e l l - k n o w n theorem 0 i s i n t e r i o r t o I ^(B°) ( t o p o l o g i c a l l y ) . T h i s p r o v e s t h a t I i s c o n t i n u o u s . 0.20 C o r o l l a r y : I : E —> 1(E) i s a homeomorphism f r o m t h e F r e c h e t s p a t e E o n t o i t s image I f E ] i n [ ( E * , s ) * , s ] w*-compactriess As an easy consequence of t h e B a n a c h - A l a o g l u theorem 10 [6; theorem 3.15 page 66] we s t a t e t h e f o l l o w i n g p r o p o s i t i o n . 0.21 P r o p o s i t i o n : L e t E be a Banach s p a c e . L e t K £ E* be noicm bounded. Then K.. = w* - C£(K) i s w*-compact. 11 C h a p t e r 1 THE CHIPPING LEMMA AND THE KREIN-MILMAN  EXTENSION THEOREM I n t h i s c h a p t e r , an e x t e n s i o n o f t h e K r e i n - M i l m a n t h e o r e m . i s o b t a i n e d by means o f the C h i p p i n g Lemma ( I . Namioka [1]). A p r o o f o f t h e R y l l -N a r d z e w s k i f i x e d p o i n t theorem ( I . Namioka and E. A s p l u n d [2]), t o w h i c h t h e c h i p p i n g lemma was o r i g i n a l l y a p p l i e d , i s p r e s e n t e d , and v a r i o u s a p p l i c a t i o n s o f t h e R y l l - N a r d z e w s k i f i x e d p o i n t theorem a r e s k e t c h e d t o p r o v i d e some m o t i v a t i o n f o r t h i s a r e a o f r e s e a r c h . 1.1 D e f i n i t i o n : An a f f i n e map T fro m a convex s e t K i n t o i t s e l f i s a map w h i c h s a t i s f i e s T ( a x +3£y) = aTx + gTy f o r e v e r y x,y £ K and e v e r y a,g>_0,a + g = l . A s u b s e t Q o f K i s T - i n v a r i a n t i f T(Q) £ Q . F o r S a c o l l e c t i o n o f a f f i n e maps, Q i s - S - i n v a r i a n t i f and o n l y i f Q i i s T - i n v a r i a n t f o r each T I S 1.2 D e f i n i t i o n : A c o l l e c t i o n S o f a f f i n e maps fr o m K i n t o K i s a semigroup i f i t i s c l o s e d w i t h r e s p e c t t o c o m p o s i t i o n o f mappings. A semigroup S o f a f f i n e maps i s f i n i t e l y g e n e r a t e d i f a l l members o f S a r e c o m p o s i t i o n s o f a f i x e d f i n i t e s u b c o l l e d t i o n o f S 1.3 D e f i n i t i o n : L e t ( E , J ) be a l o c a l l y convex t o p o l o g i c a l v e c t o r s p a c e . L e t Q £ E and l e t S be a semigroup o f a f f i n e maps s u c h t h a t Q i s S - i n v a r i a n t . S i s J - n o n c o n t r a c t i n g on Q i f f o r e a c h d i s t i n c t p a i r x,y c Q , Q i J - C£({Tx - Ty: T £ S}) . 1.4 P r o p o s i t i o n : L e t Q £ (X,J) , and S a semigroup o f a f f i n e maps 12 from Q into Q Then S i s J-noncontracting i f and only i f f o r every d i s t i n c t x,y £ Q there e x i s t s a J-continuous pseudo-norm p on Q such that Proof: S i s J-noncontracting. Thus for every d i s t i n c t x,y € Q there e x i s t s a balanced, convex, absorbing 0-neighbourhood V £ X such that Tx - Ty ^  V for every T € S . The Minkowsky functional u i s the required continuous pseudo-norm. Conversely assume that f o r every d i s t i n c t x,y £ Q there i s a J-continuous pseudo-norm p such that <5 = inf{p(Tx - Ty)} > 0 . Let V = {x £ Q: p(x) < 6} . Then V i s a O-neighbourhood i n X. ., and 1.5 Theorem: (Ryll-Nardzewski) [I.Namioka, E. Asplund; 2] Let (E, J) be a l o c a l l y convex hausdorff t o p o l o g i c a l vector space. Let Q S. E be non-empty, convex and weakly compact. Let S be a J-non-contracting semigroup of weakly continuous a f f i n e maps of Q into Q Then there i s a point z € Q such that Tz = z f o r a l l T £ S . (That i s , z i s a common f i x e d point of S on Q ). Before presenting the proof of t h i s theorem, we examine i t s p r i n c i -p a l a p p l i c a t i o n to the existance of a l e f t i n v a r i a n t mean on W(G) - the set of weakly almost p e r i o d i c functions from a l o c a l l y compact group G into C (F. Greenleaf [12; chapter 3]). 1.6 D e f i n i t i o n : Let G be a l o c a l l y compact group. B(G) i s the space of a l l bounded complex-valued functions on G equipped with the supnorm i n f {p(Tx - Ty)} > 0 T€S TSS for every T € S 13 f f . f l ^ . CB(G) i s t h e subspace o f c o n t i n u o u s f u n c t i o n s . 1.7 D e f i n i t i o n : . L e t G be a l o c a l l y compact group. L e t f £ CB(G) The l e f t o r b i t o f f i s d e f i n i e d by L 0 ( f ) = { x f : x £ G} , where * -1 x f ( y ) = f ( x g) , f o r e v e r y g £ G . 1.8 D e f i n i t i o n : L e t G be a l o c a l l y compact group, f « CB(G) i s weak-l y a l m o s t p e r i o d i c i f and o n l y i f L 0 ( f ) i s r e l a t i v e l y w e a k l y compact i n CB(G) . (That i s t h e weak c l o s u r e o f L 0 ( f ) i s w e a k l y compact i n CB(G) .) The space o f a l l s u c h f u n t i o n s i s denoted by W(G), 1.9 D e f i n i t i o n : A l i n e a r f u n c t i o n a l m on W(G) i s a mean i f m(f) = mf f o r a l l f £- W(G) . f d e n o t e s t h e c o n j u g a t e f f u n c t i o n t o i f . . and i n f { f ( x ) } <_.m(f) <_ sup { f ( x ) } f o r a l l r e a l v a l u e d x£G xcG f e W(G) f W ( G ) I f f u r t h e r m o r e m ( x f ) = m(f) f o r each x £ G , and f i n W(G) t h e n m i s s a i d t o be a l e f t i n v a r i a n t mean on W(G) 1.10 Theorem: [12; pages 38-40] L e t G be a l o c a l l y compact group. Then W(G) has a l e f t i n -v a r i a n t mean. S k e t c h o f P r o o f : L e t (§(f) be t h e w e a k l y c l o s e d conves h u l l o f L 0 ( f ) , where f £ W(G) . Then Q ( f ) i s non-empty convex and w e a k l y compact. D e f i n e L : Q ( f ) —> Q ( f ) by L (h) = x - 1 h f o r x £ G . Then L X X X i s an a f f i n e map. A l s o S = { L x : x £ G} i s n o r m - n o n c o n t r a c t i n g . I n -deed, i f f 1 ^ f 2 t h e n | f - f 2 || > 0 , and hence 0 ( Cl{Ly.£1 -14 = C£{L x(f 1 - f 2 ) > x £ G since inf - f 2) || = inf || f x - > 0 . x&G xtG Since S is a semigroup of weakly continuous maps which are norm non-contracting, the Ryll-Nardzewski fixed point theorem yeilds some h^ £ Q(f) such that I^ Ch^ ) = h^ for every x e G . Then h^ is a constant function on G , since h^(xg) = h^(g) for a l l x, g e G -1 - I i Hence for g = x one gets h^(e) = h£,ig ;) for a l l g £ G so h^ takes the constant value h^(e) on G A detailed proof that; h^ is the unique fixed point of S in Q(f) ; that the map m: W(G) —>- C which assigns to each f e. W(G) the value of the constant function h^ , is linear; and that inf f(x) xeG _< m(f) <_ sup f(x) for a l l real valued f £ W(f) , is shown in [12; X6.G page 39-42] for details) Given then that m is a mean, i t is clearly l e f t invariant since L0(f) = LO(xf) for a l l x £ G , thus the unique fixed point of L0(f) coincides with that of L0(f ) X We now present I. Namioka's and E. Asplunds proof of the R y l l -Nardzewski fixed point theorem. [2l]2] (Theorem 1.5) Proof: It suffices to prove the result for S a f i n i t e l y generated set of affine maps. Indeed, the assumption that S has no common fixed point, but that each f i n i t e subset of S does, leads to a contradiction as follows: Since S has no common fixed point x e Q , Tx ^ x for some T £ S That i s Q = T^{x: Tx - x / 0} . Now {x: Tx - x 4 0} = comp '({: Tx - x = 0}) i s weakly open for each T £• S , since T i s weakly continuous. Q i s weakly compact, thus Q ""J^tx" T i x ~ x ^ }^ for f i n i t e l y many T;j. £ S . This says that {T^, . ..,T } hassho common fixed point, which is a contradiction. Thus we assume that {T1, ...,Tm> is a f i n i t e 15 generating set for S Consider T 0 = T,+ ... +T vt' J. m m Q is convex, therefore i'T : Q —>• Q . Also T q is weakly continuous and affine. Thus the Markov-Kakutani (0.14) applies to T q , and there exists a fixed point x of T o o We show that X q ms the required fixed point for S Assume not: Without loss of generality we can assume that X q is a fixed point of no T. £ S ( (We simply discard those for which T.(x ) = x and — 1 • r y 1 0 0 work with the remaining J-noncontracting subsemigroup.) Since S is J-noncontracting, by proposition 1.3, there i s a J-continuous pseudo-norm p and an e > 0 such that: (1) p(TT.(x ) - T(x )) > e for every T. £ S , i = 1 ... m . r i o o o 1 Let K = J - C£.[Co({Tx.o: T S}) ] . K is weakly compact since K i s a subset of the weakly compact set Q , and K is weakly closed since i t i s J-closed and convex. Also K i s J-separable since the rational convex combinations of the countable collection {Tx : T e S} xo (S is f i n i t e l y generated) i s a countable J-dense subset of K If we further assume now that there exists a closed, convex C S K such that C ^ K , but that the p-diam(K^C) L ' E (Chipping lemma) then a contradiction can be achieved as follows: Let C be the above postulated subset of K . Now there i s some S & S such that S X q £ K >> C since K \ C is open in K . Sv = ST (x.) = ST,x + ST„x + ... + ST x X0 O o 1 o 2 o m o m 16 C i s convex, thus ST.x £ K^C for some i = 1, m . Hence 1 o p(ST.x - S ) < p-diam(K^C) < e which contradicts (1). r l o x — — o It remains to prove the chipping lemma. 1,; 1.11 Chipping lemma:. (I. Namioka, E. Asplund [2]) Let (E,J) be a locally convex, hausdorff topological vector space. Let K c E be non-empty, weakly compact, convex, and such that K i s contained in some J-separable set in E . Then for every e > 0 there is a J-closed convex C £ R such that C ^  K , and the p-diam(K^C) <_ e . Remark: In [2] K is taken to be J-separable. (outline of proof): The method of the proof consists in taking for some u € ext(K) , convex combinations. Cg. = {Au^ + (1 - X)u: 0 < r < 1, X £ [r-,1], u ± e. ext(K)'v{u}} . Then u ft since X ? 0 and u is an extreme point of K . As £ tends towards 0 , C^ tends towards K . Eventually some C^ is chosen as C Of course the set C so derived does not conform to the requirements of this lemma, since i t is neither closed nor convex. The procedure which we followiis to find a weakly open set W such that p-diam(W) <_ TT such that W contains an extreme point of K ^ Convex combination Cg. of the form Ct_='{Xx1 + (1 - X)x 2: 0 < ,f'< 1 , A £ [ r , l ] , X ; L £ j - C£[C 0 (D >• W) ], x 2 £ J - C£[CQ(D 0 N)]} where D ?.ls the weak closure of ext(K) w i l l satisfy this lemma for some sufficiently small .i . (proof of chipping lemma) ; , Let S = {x: p(x) <_ T-} / S is convex. 1 7 ( 1 ) S i n c e p i s J - c o n t i n u o u s , S i s J - c l o s e d . T h u s S i s w e a k l y c l o s e d . N e x t , ( 2 ) s i n c e p i s J - c o n t i n u o u s c o u n t a b l y m a n y t r a n s l a t e s o f S c o v e r K . T h i s i s t r u e s i n c e K i s c o n t a i n e d i n a J - s e p a r a b l e s e t , a n d J - i n t ( S ) T 0 S l a t e * ' S - " M £ w e a k l y " c T c 5 s " e c i , * € n § s € E a n s l » S t e s a c o v e r . - - D - = - J ^ C £ [ e x t ( K ) ] B u t D i s a w e a k l y c l o s e d s u b s e t o f t h e w e a k l y c o m p a c t s e t K T h e r e f o r e D i s w e a k l y c o m p a c t . T h u s ( D , J W ) i s a B a i r e S p a c e , a n d h e n c e s e c o n d c a t e g o r y i n i t s e l f . ( r e m a r k 0 . 1 5 ) . T h e r e f o r e t h e r e i s a k £ K s u c h t h a t J - i n t ( k + S ) fl D ^ 0 . w C l e a r l y t h e n e x t ( K ) C\ W 4 0 w h e r e W = J - i n t ( k + S ) . L e t w u € . e x t ( K ) f> W . L e t C _ • = ' { A x . . + ( 1 - A ) x 0 : 0 < ry< 1 , A £ [ r , , l ] , x . . £ J - C £ [ C ( D > - W ) ] , x „ e J-c£[c ( D n W ) ] } . z o W e s h o w t h a t C ^ i s J - c l o s e d , c o n v e x , t h a t C r . ^  K , a n d t h a t C r c a n b e m a d e p - a r b i t r a r i l y s m a l l b y c h o o s i n g r s u f f i c i e n t l y s m a l l . C o n s i d e r t h e j o i n t l y c o n t i n u o u s m a p f r ( . x ^ J - e | [ C X ; ( D ^ W ) ; ] X J - C £ [ C O ( D O W ) ] X [ r , l ] K d e f i n e d b y f r ( x ^ , x 2 , A ) = A x ^ + ( 1 - A ) x 2 • C l e a r l y t h e i m a g e o f t h i s m a p i n K i s C , w h i c h i s t h u s s h o w n t o b e J - c l o s e d s i n c e t h e d o m a i n i s c o m p a c t . C r i s c o n v e x , s i n c e i f a ^ g , Y [ £ , 1 ] , x ^ , y ^ £ J - C £ [ C Q ( D V W ) ] , a n d x 2 , y 2 £ J - C £ [ C Q ( D f\ W ) ] t h e n y(ax1.+ ( 1 - a ) x 2 ; + (qx^+gX^T- oOx^ ) + ( 1 -• * ) i ( | y 1 w , * e i ( « l " - 6 - 8 > y £ ) - = Y * " z j ; f c ( 1 - . r . ) z 2 w h e r e §5= 3[£,Yl-+s^ae([;r,[l]- X$ -5r©-"• [.f.,i-]n^slttceYYll'.3 ' g - Y3 + uYa < B - Y3 + T £ 3 + y\<k -$3)<<gg + ( 1 - B B ) = 1 , a n d 18 - y& + ya. >_ $ - By + Y R L 3(1 - y) + Y R : L (1 - Y ) R + Y ^ = R •) , ^ A Y + i I« z-, = 7, ..„ , x, + 1 - and 1 B - Y B + ya 1 [ 3 - Y B + ya-yi y z = T ~ Ta x . 2 1 - 3 + yp - ya 2 X Y ~ Y A 1 - B + Y3 - Y A 2 and the coefficients of x^ and c a n ^e shown to be in [0,1] hence z]L 6. J-C£(CQ(D N W ) ) and z 2 £ J-C£(C0(D fl W ) ) . u (the extreme point of K found in W ) ft C , since i f u € C r then u would be an extreme point of C r . By theorem 0.12 this would imply u e. D N W contradicting u £ W . This shows C ^ K . Finally we show that the p-diam of i s arbi t r a r i l y small for small £ . Consider f „ defined above with r = 0 . The image of f G is J-closed and convex and contains a l l of the extreme points of K , hence i t equals K (Krein-Milman theorem). That is every x £ K can be written x = Ax + (1 - A)x2,A <= [0,1], ^ J-Ct[C 0(D - W ) ] x 2 £ J-C£[C0(D H W ) ] . Consequently for any y e. K^C£'0r / 0) y = + (1 - A)x 2 A e [0, r) Therefore p(y - x 2) = Ap(x^ - x 2) £ where C-d = p-diam(K) . But, (3) p i s J-continuous, therefore {x: p(x) <_ 1} i s weakly open and is weakly compact, hence covered by f i n i t e l y many translates of {x: p(x) <_ 1} . That i s C/d = p-diam(K) < » . now x 2 e. J-C£[C0(D fl W)] which has p-diam <_— thus p-diam (K^C r) " y ^ W c ^ P ^ l " ^ ^ - y ^ y ^ s c / P ^ l " X 2 ) + p ( x 2 " X 2 ? ) + + P(7^ ]"3'(^ )i]>}.e< | 2-tx2fd JvC(wRere x^ })x| 2 S SMcl[CQ (DH W ) ] ) 19 C = C £ s a t i s f i e s t h e c h i p p i n g lemma, and c o m p l e t e s t h e p r o o f of t h e R y l l - N a r d z e w s k i f i x e d p o i n t theorem. 1.12 Remark: D e f i n i t i o n : A s u b s e t K o f a Banach space E i s c a l l e d d e n t a b l e i f f o r each e > 0 t h e r e i s a u e K s u c h t h a t u C£[C D(K v { y : | u - y | _< e})] . u e K i s a d e n t i n g p o i n t o f K i f u i c£[C 0(Kx{y: ||u - y|| <_ e})] f o r each e»> 0 . S i n c e t h e s e t W i n t h e p r o o f i i d e f i n e d as a t r a n s l a t e o f {x: p ( x ) <_ I"} and s i n c e u £ Qt[CD(K xW) ] we might say t h a t K i s " p - d e n t a b l e " . S i n c e the p o i n t u € W was dependent on e we a r e n o t e n t i t l e d t o denote u aas a " p - d e n t i n g p o i n t " . (More on d e n t i n g p o i n t s i n s e q u e l ) . A s l i g h t m o d i f i c a t i o n o f t h e c h i p p i n g Lemma l e a d s i n t o a n o t h e r paper by Namioka [1], where t h e lemma forms one o f t h e two b a s i c t e c h n i -c a l arguments. The main t h r u s t o f Neighbourhoods o f Extreme p o i n t s [1] i s towards an e x t e n s i o n o f t h e K r e i n - M i l m a n theorem. L e t K be a compact, convex s u b s e t o f some h a u s d o r f f t o p o l o g i c a l v e c t o r space ( E , J ) . T h i s s t r o n g e r v e r s i o n i s a c h i e v e d by d e t e r m i n i n g a dense s u b s e t o f s p e c i a l p o i n t s o f e x t ( K ) . The c l o s e d convex h u l l o f t h i s s u b s e t i s c l e a r l y , a g a i n K . S i n c e the pseudo-norm p o f t h e c h i p p i n g lemma i s J - c o n t i n u o u s , t h e t o p o l o g y on .(K,5 ) ' w h i c h p g e n e r a t e s on K s a t i s f i e s t h a t each Jp-open s e t c o n t a i n s a J-open s e t . Thus, s i n c e E i s h a u s d o r f f , t h e f o l l o w i n g p r o p e r t y i s r e a d i l y seen t o h o l d f o r each x Q £ K: 20 E v e r y ( K , J ^ ) n e i g h b o u r h o o d of x 0 c o n t a i n s a ( K , J ) - o p e n s e t . I f however p i s o n l y l o w e r J - s e m i c o n t i n u o u s , t h e n f o r a g i v e n p o i n t x Q e K , the above p r o p e r t y i s n o t g u a r a n t e e d t o h o l d . T h i s can be more s u c c i n c t l y s t a t e d a s : t h e i d e n t i t y map i : (K,J) (K,Jp) may n o t be c o n t i n u o u s a t x Q . We show t h a t t h e dense s u b s e t o f s p e c i a l p o i n t s o f e x t ( K ) r e f e r e d t o above a r e p r e c i s e l y the p o i n t s o f c o n t i n u i t y of t h e i d e n t i t y map i w h i c h a r e i n e x t ( K ) S i n c e the c h i p p i n g lemma i s t o be our main t o o l i n p r o v i n g t h i s a s s e r t i o n , we b r o a d e n i t t o i n c l u d e l o w e r J e s e m i c o n t i n u o u s pseudo-norms i n s t e a d of j u s t J - c o n t i n u o u s ones.. We compensate f o r t h i s s t r e n g t h e n e d r e s u l t by s t r e n g t h e n i n g t h e s e p a r a b i l i t y c o n d i t i o n on K 1.13 P r o p o s i t i o n : L e t ( E , J ) be a l o c a l l y convex, h a u s d o r f f t o p o l o g i c a l v e c t o r s p a c e . L e t p be a l o w e r J - s e m i c o n t i n u o u s . p s e u d o - n o r m on E L e t K £ E be convex, J-compact, and su c h t h a t i t i s c o n t a i n e d i n some J p - s e p a r a b l e s e t . Then f o r each e > 0 t h e r e i s a J-compact, convex C £ K such t h a t p-diam(K ^ C) f§5.££ , b u t C f K . p r o o f : To m o d i f y t h e c h i p p i n g lemma we show t h a t a l l o f t h e s t e p s j u s t i -f i e d by t h e J - c o n t i n u i t y o f p i n t h e p r o o f can be o b t a i n e d w i t h t h e p r e s e n t h y p o t h e s i s . S i n c e the J w - c o m p a c t n e s s o f K f o l l o w s f r o m t h e J -compactness o f K , t h e r e s u l t w i l l f o l l o w . The r e l e v a n t s t e p s have been numbered ( 1 ) , (2) and (3) i n t h e p r o o f o f Theorem 1.5. (1) - S i s J - c l o s e d s i n c e p i s J - c o n t i n u o u s . --£rj S i n c e p i n o u r -present'fhyjTothes'is- i s * l o w e r J - s e m i c o n t i n u o u s , -^-S = {x: p ( x ) <_ 1} i s J -c l o s e d . S i n e s K i n our p r e s e n t h y p o t h e s i - is J., ' 21 (2) — S i n c e p i s J - c o n t i n u o u s c o u n t a b l y many t r a n s l a t e s o f S c o v e r K S i n c e K i n our p r e s e n t h y p o t h e s i s i s c o n t a i n e d i n a J p - s e p a r a b l e s e t we need o n l y t h a t J p - i n t ( S ) ^ 0 . T h i s i s c l e a r l y t r u e s i n c e S = {x: p ( x ) < J } . (3) — I d = p-diam(K) <-°° s i n c e p i s J - c o n t i n u o u s . — F o l l o w i n g i s a p r o o f o f Cd's f i n i t e n e s s b a s e d on t h e l o w e r J - s e m i -c o n t i n u i t y o f p . The p r o o f i s m o d e l l e d on t h e a b s o r p t i o n theorem [ 3 ; page 9 1 ] . L e t A = {x: p ( x ) <_ 1} . A i s convex and i t i s J - c l o s e d s i n c e p i s l o w e r J - s e m i c o n t i n u o u s . A l s o , s i n c e p i s d e f i n e d on a l l o f E , E = U nA and c o n s e q u e n t l y K = U A f\ K , where nA O K i s J - c l o s e d ntlN n J n£(N f o r e ach n £ (N . But s i n c e (K,J) i s a B a i r e s p a c e , i t i s 2nd c a t e -g ory i n i t s e l f (remark 0.16) thus t h e r e e x i s t s an N€ IN s u c h t h a t i n t ( N A f\ K) t 0 . I f n >_ N , t h e n i n t ( n A H K) = i n t ( { x : p ( x ) < n} A K) S i n t ( { x : p ( x ) <_' N} f\ K) = i n t ( N A H K) + 0' . L e t U be a J-open 0-neighbourhood and y £ K s u c h t h a t 0 t (y + U) 0 K £ i n t ( n A H K) f o r e v e r y n> >_ N . Now E i s a l o c a l l y convex space and K i s J-bounded, t h e r e f o r e K K i s a l s o J-bounded. Hence we can f i n d b £ (0,1) s u c h t h a b ( K ^ K ) £ U , f r o m w h i c h we g e t t h a t nA 2i (y + U) V\ K 2 [y + b ( K ^ K ) ] H K . K i s convex, thus bK + (1 - b ) y £ K . But y ( l - b) + bK = y + b K - b y ^ y + b ( K - K) , t h e r e f o r e y ( l - b) + bK £ y + b ( K ^ K ) n K £ nA f o r e v e r y n >_ N L e t p ( y ) = s . Then p ( ~ y ) = 1 s o ~Y & A w h i c h i s t o say y e sA . T h e r e f o r e - y ( l - b) £ sA s i n c e 1 - b < 1 . T h i s shows t h a t rA 2 j [ y ( l - b) + bK] - - | y ( l - b) 2 -|bK (A i s convex) f o r each 22 r >.m = max(s,N) Thus K £ A i e p-diam(K) < ^ < » . b — b We a r e now p r e p a r e d t o examine Namioka's e x t e n s i o n of t h e K r e i n -Milman theorem [ 1 ; theorem 2.2] 1.14 Theorem: L e t ( E , J ) be a H a u s d o r f f l o c a l l y convex t o p o l o g i c a l v e c t o r s p a c e . L e t p be a l o w e r J - s e m i c o n t i n u o u s pseudo-norm on E L e t K be a J-compact, convex s u b s e t o f E s u c h t h a t K i s J p -s e p a r a b l e . Then t h e s e t o f extreme p o i n t s of K w h i c h a r e a l s o p o i n t s of c o n t i n u i t y o f t h e i d e n t i t y map i : (K,J) — • (K,Jp) i s a J-dense Gg s e t i n e x t ( K ) Remark: Namioka t a k e s (E,Jp) s e p a r a b l e . P r o o f : L e t Z be t h e s e t o f p o i n t s o f c o n t i n u i t y of t h e i d e n t i t y map i : ( K,J) —> (K,Jp) . Then f o r u £ K , u £ Z i f and o n l y i f f o r each e > 0 we can f i n d a ( K , J ) - o p e n n e i g h b o u r h o o d o f u of p-diam <_ e S e t t i n g e = ^ , and l e t t i n g n i n c r e a s e t h r o u g h IN we can r e -00 f o r m u l a t e t h i s c o n d i t i o n t o Z = fL-Bir-nwhereQrB: =i{u £ K: U i s c o n -n = l JL e n' t a i n e d i n a ( K , J ) - o p e n s e t o f p-diam <_ e} . Note t h a t B e i s the u n i o n o f a l l open s e t s i n (K,J) of p - d i a m e t e r <_ E , hence i t i s open. We must show t h a t Z j f \ e x t ( K ) i s dense i n e x t ( K ) . T h i s w i l l be a c c o m p l i s h e d by showing: (a) t h a t B £ H e x t ( K ) i s dense i n e x t ( K ) f o r each e > 0 , and (b) ( e x t ( K ) , J ) i s a B a i r e space. T h i s w i l l g i v e us t h a t Z 0 e x t ( K ) = B^ H e x t ( K ) i s a dense Gg n "subset o f ^ e x t ( K ) '. (a) L e t W be an a r b i t r a r y open s e t i n (K,J) s u c h t h a t W H e x t ( K ) 23 ^ 0 . By the c h i p p i n g lemma we know t h a t K c o n t a i n s a c l o s e d convex s u b s e t C ^ K . I n t ( K v C ) must c o n t a i n an extreme p o i n t o f K , s i n c e i f e x t ( K ) £ C and C i s c l o s e d and convex, t h e n K £ C w h i c h cannot be. I n o t h e r w ords, 0 / e x t ( K ) (\ i n t ( K \ C ) c B e f l e x t ( K ) . I t remains t o l o c a t e t h e s e t C so t h a t I t m i s s e s a p a r t o f W c o n t a i n i n g an extreme p o i n t o f K . That i s , 0 4 e x t ( K ) C\ i n t ( K s C ) £ B £ f\ W f\ e x t ( K ) . L e t S = {x: p ( x ) <_ ' • S i s J - c l o s e d , s i n c e p i s l o w e r J -s e m i c o n t i n u o u s . L e t D = J-C£[ext(K)] . D i s compact s i n c e K i s S i n c e K i s J p - s e p a r a b l e , a c o u n t a b l e c o l l e c t i o n o f t r a n s l a t e s o f S w i l l c o v e r K and hence D CO That i s , D = .U_ D 0 ( x . + S), {x.} c K , and D H ( x . + S) i s i = l l ' l — l c l o s e d f o r each i c IN . By c o r o l l a r y 0.17 o f t h e B a i r e C a t e g o r y theorem rooo . U , i n t [ D 0 ( x . + S ) ] i s dense i n D . I t f o l l o w s t h a t i n t [ D 0 ( x . i = l I I + s S ] ] 0 W ^ 0 f o r some x^ £ K , and c l e a r l y t h i s i n t e r s e c t i o n c o n t a i n s an extreme p o i n t o f K But i n t [ D n ( x i + S)]£ B £ f o r e v e r y x_^  e K . T h e r e f o r e e x t ( K ) O B £ 0 W 4 0 . (b) That ( e x t ( K ) , J ) i s a B a i r e space i s a theorem o f Choquet's [4; page 3 5 5 ] . I t i s t r a n s l a t e d and i n c l u d e d h e r e i n f o r t h e sake o f c o m p l e t e n e s s , and because o f t h e i n t e r e s t i n g t e c h n i q u e s o f t h e p r o o f . E x p l a n a t o r y comments a r e i n i t a l i c s . 1.15 Theorem: L e t E be a l o c a l l y convex s e p a r a b l e s p a c e , C a compact, convex, s u b s e t o f E, A the s e t o f extreme p o i n t s o f C Then A i s a B a i r e Space. p r o o f : F o r a l l c o n t i n u o u s l i n e a r f u n c t i o n a l s f on E , and f o r a l l r e a l numbers a , l e t us denote by Uf a ( r e s p F f } 0 1) t h e s e t o f x £ C such 24 t h a t f (x) < a ( r e s p . f (x) <_ a) The strategy of the proof is to show that Ff-;.a fOT which x e. Uf3a form a basic neighbourhood system for x if x e A Next given a sequence {Vn} of dense open sets in A and an arbit-rary open V in A 3 we embed the Vn's and V in appropriate sets in C . In particular V is embedded in U = U,f „ r J'2*a'l fov some f^ c E* 3 aj,^. ft • . Next we find a sequence {(f^* — ^ such that the Fj. q are descendinq compact, non-empty and such that F„ . , , . ,7 o r r ° J .a intersects with J n3 n V and V • , . The non-empty intersection of the {F„ a } will contain n ' ^ i 3 ai a member of each 3 and of V 3 and the result follows. L e t x e A . We w i l l f i r s t show t h a t t h e s e t of F' f o r w h i c h f ,a x £ » a f o 1 1 1 1 a ^ a s^- c n e i g h b o u r h o o d s y s t e m o f x . By t h e Hahn-Banach Theorem, t h e i n t e r s e c t i o n o f t h e s e F... i s {x} f ; a Indeed E* separates points on E 3 and thus if x3y e F^ a then g33 can be found such that y t F S o . 9 > 3 S i n c e the F f E.v a r e compact, i t s u f f i c e s t o show t h a t t h e f a m i l y o f t h e F - 9 f o r w h i c h x £ U f i s a d e s c e n d i n g n e t . L e t F n , f„, a-, ou be such t h a t x £ tL. n U r . L e t C., C„ be t h e complements 2 f l » a l f 2 ' a 2 1 2 of U r , Uj. i n C S i n c e C„ and C„ a r e compact, t h e convex f ^ a ^ f 2 , a 2 1 2 h u l l o f U C 2 i s compact and x ^ because x i s an e x t i p o i n t . If x £ C^ then by theorem 6.12 x e C^ U C^ since C^ U is compact and x € A . This contradicts x £ U„ ' fl U„ ' ;reme 25 Thus t h e r e e x i s t s a l i n e a r c o n t i n u o u s f u n c i t o n a l f on E , and a r e a l number a such t h a t f (x) < a , and f (y) Ss a f o r y £ separation theorem for convex sets So Fj. d o e s n ' t i n t e r s e c t C, o r C„ . That i s F " £ f , a 1 2 f , a ~ f ^ ' 0 ^ f „ ,a„ and x e 2 2 f ,a L e t ^ v n ^ he a sequence of open dense s u b s e t s o f A . I t i s r e -q u i r e d t o p r o v e t h a t O V n i s dense i n A . That i s i t i n t e r s e c t s e v e r y non-empty open s u b s e t V of A L e t {^^} , U , be open sub-s e t s o f C w i t h U dense i n C , s u c h t h a t U R A = V , U H A = V . n ' n 1 n ' One can assume t h e U 's and V 's a r e d e s c e n d i n g and t h a t U i s o f n n b the f o r m U,. f i » ° i • Let V ' =.H V. and work with {V '} instead of {V } . n t<m % n J n T h i s b e i n g t h e c a s e , we w i l l p r o v e t h e e x i s t e n c e o f ( f ^ , a ^ ) , { f ^ , a ^ ) , ... w i t h t h e f o l l o w i n g p r o p e r t i e s : F. Q. U,. n U ,, and n+1 n+1 -_n n U,. H A ^ 0 f o r each n £ IN . f , a n n „ We a l r e a d y . h a v e ( f ,ot ) . Suppose we have f o u n d ( f . ,a ) , ... ( f ,a ) 1 1 i I n n •There - exifefcseafi-.* x t e Un x f l A\ f\ U , 1 \. Hence .U-l.an f l U . . . i - f ,a • i . ,an+1 n-t: f , a n+1 n n n J n n n i s a n e i g h b o u r h o o d of x i n C , t h u s t h e r e i s ^ n + i ' a n + i ^ s u c h t h a t f .,,a , n f ,i,au+1 f ,a n+1 n+1' n+1 n+1' n+1 n' n ^f a) ^s a neighbourhood system of x in C . S i n c e x e A , we get t h a t U f P\ A ^ $ and we can c o n t i n u e n + l ' a n + l i n d u c t i v e l y , The F^ a r e d e s c e n d i n g , noncempty, compact. T h e r e f o r e t h e y have n' n a non-empty i n t e r s e c t i o n F . F £ U,. and F £ R U . F i n a l l y F r ^ , r a ^ n 26 Thus t h e r e e x i s t s a l i n e a r c o n t i n u o u s f u n c i t o n a l f on E , and a r e a l number a such t h a t f ( x ) < a , and f ( y ) > a f o r y £ separation theorem for convex sets So F.. d o e s n ' t i n t e r s e c t C, o r C« . That i s F,. ^ U„ f ,a 1 2 f ,a ~ f 2 . ' a l ^ * r o s a „ and x € U, 2 ' 2 f,ot L e t { v n ^ ke a sequence o f open dense s u b s e t s o f A . I t i s r e -q u i r e d t o p r o v e t h a t R i s dense i n A . That i s i t i n t e r s e c t s e v e r y non-empty open s u b s e t V o f A L e t tU } , U , be open s u b -s e t s o f C w i t h U dense i n C , s u c h t h a t U H A = V , u n A = V . n n 1 n One can assume the U 's and V 's a r e d e s c e n d i n g and t h a t U i s o f n n t h e f o r m U. f 1 , a 1 . Let V ' =.H V. and work with {V '} instead of {V } . n %<m % n J n T h i s b e i n g the c a s e , we w i l l p r o v e t h e e x i s t e n c e o f ( f ^ , a ^ ) , ( f 2 . o ^ ) , ... w i t h t h e f o l l o w i n g p r o p e r t i e s : F- £. U,. A D , , and f ,. ,a ,. f ,a n+1 n+1 n+1 n n U c H A ± 0 f o r each n « FN . f ,a n n We a l r e a d y have ( f ^ , o t ^ ) . Suppose we have f o u n d ( f ^ , a ^ ) , . . . ( f ,a ) There e x i s t s an x € U. D A f i l l . Hence Vc f\U , n f ,a n+1 f ,a n+1 n n n n i s a n e i g h b o u r h o o d o f x i n C , t h u s t h e r e i s ^ n + i » a n + i ^ s u c h t h a t f ,1>A ,1 f .1»a , n f ,a n+1 n+1 n+1 n+1 n+1 n n {Up ^} is a neighbourhood system of x in C . S i n c e x 6 A , we g e t t h a t U f (\ A £ 0 and we can c o n t i n u e n + l ' V h l i n d u c t i v e l y , The F^ a r e d e s c e n d i n g , non-empty, compact. T h e r e f o r e t h e y have n ' a n a non-empty i n t e r s e c t i o n F . F £ U- and F £ 0. U . F i n a l l y F f l » ° l n 27 i s compact, convex, and i t s complement i n C i s convex. A s i m p l e lemma p r o v e s t h a t F c o n t a i n s a t l e a s t one extreme p o i n t y o f C ( I n a l l F c o n t a i n s an extreme p o i n t x . I f x i s an extreme p o i n t o f C , we're done. I f n o t , l e t 6 be a s t r a i g h t l i n e t h r o u g h x and s u c h t h a t x i s an i n t e r i o r p o i n t o f C 0 6 . Then one shows t h a t one o f t h e end p o i n t s o f F H 8 i s an extreme p o i n t of C ) . The extreme point of C in F D 6 can be found in a Un . y £ A f l U H V f o r each n c FN,., and y e A f l D c = V . l ' a l T h i s c o n c l u d e s Choquet's theorem and theorem 1.14. 28, C h a p t e r 2 APPLICATIONS Many a p p l i c a t i o n s of t h e K r e i n - M i l m a n e x t e n s i o n theorem 1.14 o c c u r by a s s o c i a t i n g t h e l o w e r J - s e m i c o n t i n u o u s pseudo norm p on t h e l o c a l l y convex space ( E , J ) w i t h a n o t h e r t o p o l o g y on E . We b e g i n t h i s seed: t i o n on a p p l i c a t i o n s by i n v e s t i g a t i n g t h e s e t o p o l o g i e s on E and r e f o r m -u l a t i n g theorem 1.14 t o f a c i l i t a t e t h e f u r t h e r a p p l i c a t i o n s . The t o p o l o g y w i t h w h i c h p can be most d i r e c t l y a s s o c i a t e d i s a norm t o p o l o g y — namely i n t h e c a s e t h a t p i s i t s e l f a norm.. The p r o b i lem w i t h c h o o s i n g ( E , J ) a Banach space w i t h norm p i s t h a t t h e n p i s J - c o n t i n u o u s , and hence t h e i d e n t i t y map i : ( E j J ^ ) —> (E,J) i s everywhere c o n t i n u o u s making theorem 1.14 t r i v i a l . C o n s i d e r however J t o be t h e weak t o p o l o g y on a normed space ( E , J ^ ) . 2.1 Lemma: I n a normed space ( E , J ^ ) t h e norm p i s l o w e r w e a k l y s e m i -c o n t i n u o u s . F u r t h e r m o r e i f E i s i n f i n i t e d i m e n s i o n a l t h e n t h e norm p i s n o t w e a k l y c o n t i n u o u s . P r o o f : L e t S = {x: p ( x ) <_ 1} To show S i s w e a k l y c l o s e d c o n s i d e r a w e a k l y c o n v e r g e n t sequence x > x f o r w h i c h each x £ S n n I f x S , t h e n by t h e Hahn-Banach theorem, t h e r e i s an f £ E* s u c h t h a t f (x) > 1 and f (y) <^  1 f o r e v e r y y £ S . But x^ — > x w e a k l y means p r e c i s e l y t h a t f ( x n ) —* f ( x ) f ° r each f £ E* w h i c h i s a c o n t r a d i c t i o n . Hence x e S Next assuming t h a t E i s i n f i n i t e d i m e n s i o n a l , we show t h a t p i s n o t w e a k l y c o n t i n u o u s . {V = {x: | f . ( x ) | < r . , l ^ < _ ' i <_n,f. € E*}} i s a 29 l o c a l base f o r t h e weak t o p o l o g y on E . Thus e v e r y weak O-neighbourhood c o n t a i n s a subspace of t h e f o r m N = {x: f ^ ( x ) = 0 , 1 <_ i <^  n} . But N i s t h e n u l l space o f t h e map f r o m E i n t o IR n w h i c h t a k e s an element x £ E t o ( f 1 ( x ) , f . ( x ) , ..., f ( x ) ) £ R n . 1 2. n dim (E) <^  n + dim(N) ,, t h e r e f o r e dim(N) = °° T h i s shows the N (hence V) i s n o t p-bounded and p i s n o t c o n -t i n u o u s . T h i s g i v e s t h e f i r s t c o r o l l a r y t o theorem 1.14. 2.2 Theorem: L e t ( E , J ) be a normed s p a c e , p i t s norm. L e t K £ E be convex, w e a k l y compact, and norm s e p a r a b l e . L e t Z be t h e p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i : ( K , J ) —> ( K , J ) . Then Z O e x t ( K ) w p i s w e a k l y dense i n e x t ( K ) . Hence K = J w-C£[C Q(Z H e x t ( K ) ] 2.3 Remark: T h i s r e s u l t a l s o f o l l o w s f r o m t h e work of Joram L i n d e n s t r a u s s [5; t h e o r e m 4 ] . Theorem: E v e r y w e a k l y compact, convex s u b s e t o f a s e p a r a b l e Banach space i s the c l o s e d convex h u l l o f i t s s t r o n g l y e x p o s e d p p o i n t s , ( s i n c e s t r o n g l y exposed p o i n t s a r e (a) p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i : (K,J„) —> ( K , J ) and (b) extreme p o i n t s o f K w p D e f i n i t i o n : a p o i n t x i n a convex s u b s e t K o f a Banach-space E i s a s t r o n g l y exposed p o i n t o f K i f and o n l y i f t h e r e i s an f £ E* such t h a t ( i ) f (y) < f (x) f o r each y e. K , y ^ x and ( i i ) ^ ( x n ^ — y f 0 0 i m p l i e s ||x n - x|| —> 0 (a) x i s a p o i n t o f c o n t i n u i t y o f t h e i d e n t i t y map (K,JW) —> ( K , J ) w p means t h a t I x - x l l —>• 0 whenever x —*• x w e a k l y . S i n c e x —>• x 11 n 11 n n w e a k l y i s e q u i v a l e n t t o f ( x ^ ) — • f ( x ) f o r e v e r y f C E* we g e t t h a t a l l 30 s t r o n g l y exposed p o i n t s »f K a r e p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i (b) That a l l s t r o n g l y exposed p o i n t s a r e extreme i s c l e a r , s i n c e i f x i s s t r o n g l y exposed and x = Ax^ + (1 - A ) x 2 f o r A £ [0,1] , t h e n f ( x ) = f ( A x ^ + (1 - A ) x 2 ) = A f ( x ^ ) + (1 - A ) f ( x 2 ) w h i c h can o n l y o c c u r i f x = x^ o r x 2 ( ( I f x^ ^ x t h e n f ( x ^ ) < f ( x ) ) L e t E be a normed s p a c e . Then E* , i t s c o n t i n u o u s d u a l i s a l s o a normed s p a c e . A n a l a g o u s l y t o lemma 2.1 we have t h a t t h e norm on E* i s l o w e r w * - s e m i c o n t i n u o u s , and n o t w * - c o n t i n u o u s . Thus: 2.4 Theorem: [ 1 ; theorem 3.2] L e t K be a norm s e p a r a b l e , w*-compact convex s u b s e t o f E* , where E i s a normed s p a c e . L e t Z be t h e s e t o f p o i n t s of c o n t i n u i t y of t h e i d e n t i t y map i : (E*,w*) —*- ( E * , norm) . Then Z Q e x t ( K ) i s w*-dense i n e x t ( K ) , hence K = w*-C£[C 0(Z O e x t ( K ) ) ] We a b s t r a c t f r o m t h e f o r e g o i n g , t h e f o l l o w i n g 2.5 Theorem: L e t E be a normed s p a c e , p i t s norm. L e t J be a l o c a l l y convex t o p o l o g y on E such t h a t p i s l o w e r J - s e m i c o n t i n u o u s . L e t K be a J-compact, convex s u b s e t o f E , s u c h t h a t K i s norm-s e p a r a b l e . Then Z f l e x t ( K ) i s J-dense i n ext'(K) , where Z i s t h e s e t o f p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i : (K,J) —> (K,J^) The n e x t o b v i o u s s p a c e s t o l o o k a t a r e l o c a l l y convex pseudo-m e t r i z a b l e s p a c e s . I f ( E , J ^ ) i s a l o c a l l y convex p s e u d o - m e t r i z a b l e s p a c e , t h e n an i n v a r i a n t p s e u d o - m e t r i c Cdl can be chosen so t h a t f o r 31 A = {x: d ( x , o ) < n} , {y. } _T i s a f a m i l y o f pseudo-norms on E n >• » ' _ ' A n ne IN w h i c h d e t e r m i n e s (see d e f i n i t i o n 0 .5). A s i m p l e d e v i c e e x t e n d s theorem 1.14 t o p s e u d o - m e t r i c s p a c e s . 2.6 Theorem: [ 1 ; theorem 2 . 3 ] . L e t ( E , J ^ ) be a l o c a l l y convex p s e u d o - m e t r i c s p a c e . L e t J 2 be a n o t h e r t o p o l o g y on E such t h a t ( E , J ^ ) i s h a u s d o r f f . L e t {p } be a sequence o f J . - c o n t i n u o u s , l o w e r J 0 - s e m i c o n t i n u o u s pseudo-n n e l N ^ 1 2 norms on E w h i c h d e t e r m i n e s . L e t K be a ^ - c o m p a c t , convex J ^ - s e p a r a b l e s u b s e t o f E . L e t Z be t h e s e t of p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i : ( E , J 2 ) ( E , ^ ) . Then Z H e x t ( K ) i s a 1^-dense s u b s e t o f c e x t ( K ) . Hence K = J 2 - C£[C 0(Z H e x t ( K ) ) ] . P r o o f : C o n s i d e r E w i t h t h e t o p o l o g y g e n e r a t e d by p^ , w i t h J 2 as a second t o p o l o g y on E , such t h a t p^ i s l o w e r J 2 - s e m i c o n t i n u o u s . L e t Z n be t h e s e t o f p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i : (K,J„) —*- ( K , J ) . Then by theorem 1.14 Z H e x t ( K ) i s a J 0 -z p n ^ r n dense G^ s u b s e t o f e x t ( K ) . By theorem 1.15 of Choquet ( e x t ( K ) , J 2 ) i s a B a i r e s p a c e . Thus Z 0 e x t ( K ) = 0 ^ 7 0 e x t ( K ) i s a J„-dense r n e f N n 2 s u b s e t o f e x t ( K ) . A l s o t h i s i n t e r s e c t i o n i s G r , s i n c e G 'ness i s o '0' c l o s e d under c o u n t a b l e i n t e r s e c t i o n s . I n c l u d e d i n the d i v e r s e a p p l i c a t i o n s w h i c h we c o v e r o f t h e f o r e g o i n g t h e o r y a r e t h a t : each bounded s u b s e t o f a s e p a r a b l e d u a l Banach space i s dent--a b l e , and t h a t each c l o s e d convex, bounded ( n o t n e c e s s a r i l y compact) sub-s e t o f a F r e c h e t space whose second d u a l i s s e p a r a b l e r e l a t i v e t o i t s s t r o n g t o p o l o g y i s t h e c l o s e d convex h u l l o f i t s extreme p o i n t s . We co n -c l u d e w i t h a s l i g h t g e n e r a l i z a t i o n o f t h e R y l l - N a r d z e w s k i f i x e d p o i n t 32 theorem, a l s o due t o Namioka. [ 1 ; theorem 3.7] The f o l l o w i n g lemma g i v e s a s l i g h t l y s t r o n g e r v e r s i o n o f theorem 2.4. 2.7 Lemma: L e t E be a Banach space s u c h t h a t E* i s s e p a r a b l e . L e t K £ E* be bounded, n o r m - c l o s e d and convex. L e t = w*-C£(K) . Then K O e x t ( K ^ ) , w h i c h i s c l e a r l y c o n t a i n e d i n e x t ( K ) i s w*-dense i n ext ( K ) . P r o o f : By p r o p o s i t i o n 0.20 we g e t t h a t i s w*-compact. Thus theorem 2.4 a p p l i e s t o c (E*,w*) . That i s , Z H e x t ^ ) i s w*-dense i n e x t ( K ^ ) where Z i s the s e t of p o i n t s o f c o n t i n u i t y o f t h e i d e n t i t y map i : (K,w*) —> (K^, norm) . We show t h a t Z £ K w h i c h c o m p l e t e s t h e p r o o f . L e t z € Z . K i s w*-dense i n , t h e r e f o r e we can f i n d a n e t { x a } on K w h i c h c o n v e r g e s w* t o z . That i s , f o r each w*-open n e i g h b o u r h o o d U o f z , t h e r e i s an a 0 s u c h t h a t f o r each a > aQ , (a i n t h e d i r e c t e d i n d e x s e t I ) , x ^ £ U . But z £ Z means t h a t each E - b a l l about z c o n t a i n s a w*-open n e i g h b o u r h o o d U . Thus x a tends t o z i n norm. S i n c e K i s norm c l o s e d , z € K 2.8 Theorem: [ 1 ; c o r o l l a r y 3 . 4 ] . L e t E be a Banach s p a c e , s u c h t h a t E* i s s e p a r a b l e . Then each norm c l o s e d , convex bounded s u b s e t o f E* i s t h e norm c l o s e d convex h u l l o f i t s extreme p o i n t s . P r o o f : L e t = w*-C£(K) where K i s norm c l o s e d , bounded and convex i n E* . E x t ( K ^ ) <fi 0 , t h u s as i n lemma 2.7 we g e t t h a t N K H e x t O O £ e x t ( K ) . We show t h a t fthissisnsuf f i c i e n t t o p r o v e t h a t 33 K = C£tC Q(ext(K))] , f o l l o w i n g a p r o o f by R i c h a r d B o u r g i n as p r e s e n t e d by N. T. Peck i n [ 7 ; lemma 1 ] , (and i n a w r i t t e n c ommunication f r o m I . Namioka)• Lemma 2.9: L e t E be a l o c a l l y convex s p a c e . Then e v e r y c l o s e d , bounded, convex s u b s e t of E has an extreme p o i n t i f and o n l y i f e v e r y c l o s e d , bounded, convex s u b s e t o f E i s t h e c l o s e d convex h u l l o f i t s extreme p o i n t s . P r o o f : Assume t h a t t h e n o n - t r i v i a l o f t h e i m p l i c a t i o n s i s f a l s e . Then t h e r e i s a c l o s e d , bounded,cconvex s e t C £ E s u c h t h a t C D = C£[C 0(ext(C))] £ C . L e t y 6 C s C Q . Then by t h e s e p a r a t i o n theorem f o r convex s e t s [6; theorem 3.4, page 58], t h e r e i s an f £ E* and Be.HR s u c h t h a t f (c) < 3 <_ f ( y ) f o r e v e r y c £ C Q . That i s = {c e C: f (e) >_ g} 0 , and ^ 0 C Q = 0 . Now D = {x £ C: f ( x ) = g} ^ 0 , and i t i s c l o s e d , bounded and convex. By o u r h y p o t h e s i s , e x t ( D ) ^ 0 s a y u e x t ( D ) . C l e a r l y u £ e x t ^ ) S i n c e u f C D 2 e x t ( C ) , u = Xa + (1 - X)b f o r some X £ (0,1) and a,b £ C . S i n c e u £ e x t ( K ^ ) , one o f a,b j- . Say a ^ But t h e n b £ , s i n c e i f b j. , the n g = f (u) = X f ( a ) + (1 - X ) f ( b ) < Xg + (1 - X)g = g w h i c h cannot be. W i t h o u t l o s s o f g e n e r a l i t y , we can l e t b = a + t«(<u - a) , where t = sup{X £ fR: a + X(u - a) £ C} Indeed, s i n c e t > 1 , f (a + t ( u - a ) ) = t f ( u ) - ( t - l ) f ( a ) > ( t - l ) g = g , so a + t ( u - a) e K± . Now b ^ C 0 2 e x t ( C ) , t h e r e f o r e t h e r e a r e c, ,c„ €. C such t h a t 34 b = 1/2(2^ + , and clearly we can find c ^ c ^ £ Let p. = -r— a + - 7 — c. for 1 = 1,2 x x where 6 = 3 - f(a) > 0 and e± = f(c ) - g > 0 for i = 1,2 . Then p^, p^ £ C since C is convex and a,c^,C2 £ C • Note that f (p ^ = f (p 2) = 3 , hence p , p e D . . - t + fl 6 + E 2 B u t U = "26 + e± + e2 P l + 26 + z± + ^  P2 which contradicts that u £ ext(D) , and the proof i s complete. We next refer back to denting points as defined in Remark 1.12 following the proof of the chipping lemma. We examine the problem posed by M. Rieffel [8; question 3] namely for which spaces are a l l bounded subsets dentable. Namioka gives a partial answer in [1; theorem 3.5]. 2.10 Lemma: [1; remarks preceding theorem 3.5]. Let E be a Banach Space. Let J be a hausdorff, locally convex topology on E such that the norm is lower J-semicontinuous. Let K £ E be such that J-C£[C0(K)] is J-compact, and K is norm-separable. Then K is dentable. Proof: Let = J - C £ [ C 0 ( K ) ] . By the chipping lemma - proposition 1.135 there exists a J-closed, convex C £ K , such that C / , and the diamCK^ C ) <_ j . But clearly K ^ C ^ 0 , since i f C 2 K then C w i l l also con-tain the closed convex hull of K , namely • Let x £ K ^ C . Then clearly C 2 K ^  B (x) where B (x) i s the closed b a l l of radius e e e around x . Therefore C 2 J - C £ [ C 0 ( K > B^(x))] . That i s x 4 J - c £ [ C 0 ( K ^ B £ ( x ) ) ] . 35 2.11 Theorem: [1; theorem 3.5]. Let E be a Banach space such the E* is separable. Then each non-empty, norm-closed, convex, bounded subset of E* contains a denting point. Hence each bounded subset of E* is dentable. Proof: Let K be a norm-closed, convex, bounded subset of E* . Let K^ = w*-C£(K) , and let u £ ext(K^) be such that u has arbi t r a r i l y norm-small w*-neighbourhoods (theorem 2.4). As in the proof of lemma 2.7, u e ext(K) . But then u is a denting point of . Indeed l e t e > 0 and W a w*-neighbourhood of u such that diam(W) £ e . Then u ^ W and since u is extreme, u £ w*-C£[CD(K-^ W) ] 2 w*-C£[C0(K^x B £(u)) ] 2 norm closure of CQ(K^"«~ B £(u)) , where again, B (u) i s the closed b a l l of radius e around u . But K i s norm e bounded hence so is K^ , and is w*-closed. Thus is w*-compact, and by lemma 2.10 K is dentable. We now prove another type of generalization of the Krein-Milman theorem. [1; theorem 3.6]. 2.12 Definition: A topological vector space i s called quasi-separable i f each bounded subset i s separable. 2.13 Theorem: Let E be a Frechet Space such that (E*,s)* i s quasi-separable with respect to the strong topology. Let K£ E be closed, bounded and convex. Then K is the closed, convex hull of i t s extreme points. Proof: Let I: E —»• (E*,s)* be the evaluation map. Let 36 K± = w*-C£(I[K]). C o n s i d e r t h e b i p o l a r (K°)° o f K . (K°)° = {F £ ( E * , s ) * : | F ( f ) | <_ 1 f o r each f £ E* w h i c h s a t i s f i e s | f (x) | <_ 1 f o r each x £ K . C l e a r l y I [ K ] c (K°)° . B u t K° i s a n e i g h b o u r h o o d of 0 i n (E*,s) t h u s (K°)9 i s w*-compact i n ( E * , s ) * (Banach A l a o g l u t h e o r e m ) . Hence (K°)° i s w*-c l o s e d and so c (K°)° . That i s i s a c l o s e d s u b s e t o f a compact s e t i n a H a u s d o r f f s p a c e . T h e r e f o r e i s w*-compact. A l s o , we g e t t h a t i s s t r o n g l y bounded. L e t I be a s t r o n g 0-neighbourhood i n ( E * , s ) * . S i n c e {B°: B i s s t r o n g l y bounded i n E*} i s a l o c a l b ase f o r [ ( E * , s ) * , s ] , V 2 B° f o r some s u c h B . Now I i s c o n t i n u o u s (theorem 0.19), t h u s I'""(B°) i s a 0-neighbourhood i n E . K i s bounded i n E , t h e r e f o r e K £• n l (B°) f o r some s u f f i c i e n t l y l a r g e n £ IN . That i s I [ K ] c nB° . But B° i s w * - c l o s e d i n ( E * , s ) * , thus YL^ £ nB° £ nv . L e t K' be t h e subspace o f ( E * , s ) * g e n e r a t e d by . S i n c e ( E * , s ) * i s q u a s i - s e p a r a b l e , sKci m i s r s t r . b h g l y i s e p a r a b l e ^ a n d " m e t r i z a b l e -Iwifehc'therihducedabppdlogy.o Thusifcheor.ems?. 6 w a p p l i e s t o i n (K',s) . t w i t h s t h e d w*pt6.pg,-logy as.'the s€condrtof.ol=ogy;tonf Kiin?.<r So f o r Z = t s e t i d f - n p o . i n t a a p f e o n t i n u i - t y - o f K t h e ) i d e n t i t y map i : (K^,w*) —>- (K^,s) (1) Z 0 e x t (K.^) i s w*-dense i n e x t (K^) C o n s i d e r a n e t {xg,} l n I [ K ] w h i c h c o n v e r g e s s t r o n g l y t o F G ( E * , s ) * . I i s a homeomorphism of E o n t o ( I [ E ] , s ) , t h e r e f o r e {x^} i s a Cauchy n e t i n E . But E i s c o m p l e t e , so c o n v e r g e s t o some x e E . S i n c e K i s c l o s e d x 6 K . Thus F = I ( x ) 6 I [ K ] , and I [ K ] i s s t r o n g l y c l o s e d i n ( E * , s ) * . T h i s g i v e s , as i n t h e p r o o f of lemma 2.7, t h a t Z £ I [ K ] . Thus Z <fl ext(K^)£ e x t ( I [ K ] ) = I [ e x t ( K ) ] By (1)-'above, w * - C X [ G 0 ( I [ e x t (K) ]).] = JC^ = w*-C£(I[K]) I n v e r t i n g back t h r o u g h I - j. we get^-that weak-C£[C 0 ( e x t ( K ) ) ] = K . We c o n c l u d e w i t h a s l i g h t g e n e r a l i z a t i o n o f t h e f i r s t theorem p r o v e d The R y l l - N o r d z e w s k i F i x e d p o i n t theorem. 2.14 Theorem: [ 1 ; theorem 3 . 7 ] . L e t ( E , J ) be a l o c a l l y convex s e p a r a b l e t o p o l o g i c a l v e c t o r s p a c e . L e t Jy be a second l o c a l l y convex, h a u s d o r f f t o p o l o g y on E , s u c h t h a t J i s d e t e r m i n e d by l o w e r ^ - s e m i c o n t i n u o u s pseudo-norms p^ on E L e t Q £ E be non-empty, convex and ^ - c o m p a c t . L e t 5 be a semigroup of ^ - c o n t i n u o u s a f f i n e maps o f Q i n t o i t s e l f , s u c h t h a t S i s J -n o n c o n t r a c t i n g on Q . Then S has a common f i x e d p o i n t i n Q o u t l i n e o f p r o o f : S i s J - n o n c o n t r a c t i n g i m p l i e s t h a t f o r e v e r y d i s t i n c t p a i r x,y £ Q , t h e r e i s a J - c o n t i n u o u s pseudo-norm p on E s u c h t h a t i n f { p(Tx - Ty)} > 0 ( p r o s i t i o n 1.4). S i n c e J i s d e t e r m i n e d by a s e t TcS {p^} o f l o w e r J 2 _ s e m i c o n t i n u o u s pseudo-norms, f o r e a c h d i s t i n c t x,y p i t s e l f can be chosen t o be l o w e r J 2 ~ s e m i c o n t i n u o u s . Indeed a J-^~0 n e i g h -bourhood B w h i c h i s J 2 ~ c l o s e d can be f o u n d w i t h i n t h e p - u n i t b a l l (= {x: p ( x ) <_ 1}) . T h i s i s t r u e s i n c e t h e p^ u n i t b a l l s a r e a l o c a l base f o r J., and each i s J„-closed. i s t h e n J., - c o n t i n u o u s , l o w e r JL Z D J_ J 0 - s e m i c o n t i n u o u s , and s i n c e p < \i„ , i n f { y T I ( T x - T y ) : T «• S} >_ i n f { p ( T x - Ty) : T £ S} > 0 . I f J 0 = J , theorem 2.14 becomes theorem 1.5 w i t h a d d i t i o n a l z w h y p o t h e s i s t h a t ( E , J ) i s s e p a r a b l e . I n t h e p r e s e n t more g e n e r a l f o r m , t h e added h y p o t h e s i s i s r e q u i r e d s i n c e t h e c h i p p i n g lemma t o theorem 1.5 r e q u i r e s t h a t t h e s e t K = J-C£[C 0{T : T £ S})] be c o n t a i n e d i n a J -x o s e p a r a b l e s e t . C o n s e q u e n t l y t h i s theorem can be p r o v e d by t h e same method as theorem 1.5 w i t h p r o p o s i t i o n 1.14 r e p l a c i n g t h e c h i p p i n g lemma 1.13. 39 BIBLIOGRAPHY [ I ] I . Namioka. Neighbourhoods o f Extreme P o i n t s . I s r a e l J o u r n a l of M a t h e m a t i c s V o l . 5 Number 3, ( J u l y , 1 9 6 7 ) . [ 2 ] I . Namioka and E. A s p l u n d . A G e o m e t r i c P r o o f o f R y l l - N a r d z e w s k i ' s F i x e d P o i n t Theorem. B u l l . Amer. Math. Soc. 73 (1967) 443-445. [ 3] J . L. K e l l y , I . Namioka, e t a l . L i n e a r T o p o l o g i c a l Spaces. D. Van N o s t r a n d , P r i n c e t o n ( 1 9 6 3 ) . [ 4 ] J . D i x m i e r . L e c C * - a l g e b r e s e t L e u r s R e p r e s e n t a t i o n . G a u t h i e r - V i l l a r s , P a r i s ( 1 9 6 4 ) . [ 5] J . L i n d e n s t r e u s s . On O p e r a t o r s Which A t t a i n T h e i r Norm. I s r a e l J o u r n a l o f M a t h e m a t i c s . 1 No 3 (1963) 139-148. [ 6 ] W. R u d i n . F u n c t i o n a l A n a l y s i s , M c G r a w - H i l l Book Company. [ 7 ] N. T. Peck. Support P o i n t s i n L o c a l l y Convex Spaces. Duke Math. J o u r n a l 38[1971] 271-278. [ 8 ] M. A. R i e f f e l . D e n t a b l e Subset of Banach Spaces w i t h A p p l i c a t i o n s t o a Radon N i k o d y n Theorem. F u n c t i o n a l A n a l y s i s : P r o c e e d i n g s o f a C o n f e r e n c e H e l d a t t h e U n i v e r s i t y o f C a l i f o r n i a , I r v i n e ( 1 9 6 7 ) . Academic P r e s s . [ 9] H. H. S c h a e f e r . T o p o l o g i c a l V e c t o r Spaces. M a c M i l l a n Company, New Y o r k ( 1 9 6 6 ) . [10] F. T r e v e s . T o p o l o g i c a l V e c t o r Space D i s t r i b u t i o n s and K e r n e l s . Academic P r e s s ( 1 9 6 7 ) . [II] E. B i s h o p and R. R. P h e l p s . The Support F u n c t i o n a l s o f a Convex 40 Set. Proceedings of a Symposia in Pure Mathematics Vol VII (Convexity) Amer. Math. Soc. (1963) 27-29. [12] F. P. Greenleaf. Invariant means on Topological Groups. Van Nostrand, Reinhold (1969). [13] N. Dunford and J. Schwartz. Linear Operators. Interscience Pub-lishers inc. New York (1958). 

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