Locally Convex Spaces over Non-Archimedean Valued Fields

Abstract

Non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers are fundamental, is a fast-growing discipline widely used not just within pure mathematics, but also applied in other sciences, including physics, biology and chemistry. This book is the first to provide a comprehensive treatment of non-Archimedean locally convex spaces. The authors provide a clear exposition of the basic theory, together with complete proofs and new results from the latest research. A guide to the many illustrative examples provided, end-of-chapter notes and glossary of terms all make this book easily accessible to beginners at the graduate level, as well as specialists from a variety of disciplines.

Descriptive topology in non-archimedean function spaces C p (X , 핂). Part I

“…then by[19, Theorem 5.6.1] we conclude that E has a compact resolution. Then by Proposition 3 the space E is analytic.…”

“…Hence, by [19,Remark 4.2.17(a)], the space X is countable. If X is countable, the subspace C p (X, K) of K X is a polar metrizable LCS.…”

Stein domains in Banach algebraic geometry

“…By hypothesis F i Ñ F j is a compatoid map and since the image of compactoid subsets by bounded maps are compactoid subsets, then also F i Ñ F j F j XE is a compactoid map. By Theorem 8.1.3 (xi) of [27] this is equivalent to say that φ i,j is a compactoid map.…”

Theorems A and B for dagger quasi-Stein spaces

“…Therefore, the projective system that defines F satisfies the hypothesis of Theorem 3.13 and since F pU n q are LB spaces we can apply Theorem 3. 17 Remark 4.23. We want to emphasize the fact that Theorem 4.21 and Corollary 4.22 apply also when the base field is Archimedean (i.e.…”

“…Basic text books for the Archimedean part of the theory are so many that we cite only the one to which we will refer to later, [12] . For the non-Archimedean theory we mention [17] and [21]. Definition 2.7.…”