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

Abstract

A class of complete non-Archimedean pseudo-normed linear spaces for which the field of scalars has a trivial valuation is introduced; we call these spaces "V-spaces." V-spaces differ from the classical normed linear spaces in that the homogeneity of the norm is replaced by the requirement that llαxll = llxll for all x and all scalars α≠0; the usual triangle inequality is modified to [ Equation omitted ] and it is assumed that the norm of an element is either zero or is equal to ρⁿ for a fixed real ρ > 1 and some integer n. The concept of a "distinguished basis" in a V-space is defined. By use of a modified form of Riesz's Lemma, it is shown that every V-space admits a distinguished basis. Each element of a V-space then has a uniquely determined series expansion in terms of the elements of a given distinguished basis. An analogue of the Paley-Wiener Theorem is proved for distinguished bases. Properties of distinguished bases are exploited throughout this work. Linear and non-linear operators on V-spaces are also studied. In the usual way, a norm is defined under which the set of bounded operators is a V-space and the set of bounded linear operators is a "V-algebra." A characterization of bounded linear operators is given as well as theorems on spectral decompositions. Under certain assumptions on the expansions of x, y, A, the existence of solutions to equations of the form xz = y in V-algebras, and of the form Ax = y in arbitrary V-spaces is proved. Approximations of the solutions are obtained. A representation theorem for continuous linear functionals on a V-space is given. This representation uses an analogue of the classical inner product. Examples of V-spaces and V-algebras discussed include spaces of functions from a Hausdorff space to a normed linear space, on which the pseudo-norm characterizes the asymptotic behaviour of the functions. Some results of the theory of pure asymptotics are extended to arbitrary V-spaces.

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