Fréchet spaces of non-archimedean valued continuous functions
type: Item
instance of: scholarly article
title: Fréchet spaces of non-archimedean valued continuous functions
zbMATH Open document ID: 1236.46068
author: W. Śliwa
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-10-24T00:00:00Z
full work available at URL: http://hdl.handle.net/10593/4298
review text: This paper is an interesting contribution to the theory of non-archimedean spaces of continuous functions. A well-known result, related to the existence of orthogonal bases in the Banach setting, is extended to the locally convex case. Let \(K\) be a non-archimedean non-trivially valued field which is complete under the metric induced by its valuation. In 1968 van der Put proved that \textit{if \(X\) is a locally compact zero-dimensional space then the Banach space \(C_0(X, K)\) of all continuous functions \(X \rightarrow K\) vanishing at infinity has an orthonormal base consisting of \(K\)-valued characteristic functions of open compact subsets of \(X\).} The main theorem of this paper is an extension to the locally convex setting of the above result of van der Put. In fact, the author proves the following. Theorem. Let \(X\) be a zero-dimensional Hausdorff space, let \(C_c(X,K)\) be the locally convex space of all continuous functions from \(X\) to \(K\) with the topology of uniform convergence on the compact subsets of \(X\). Suppose \(C_c(X,K)\) is a Fréchet space. Then \(C_c(X,K)\) has an orthogonal base consisting of \(K\)-valued characteristic functions of clopen subsets of \(X\) and it is isomorphic to the product \(C(X_1,K) \times \prod_{n=1}^{\infty} C_0(X_n',K)\) of Banach spaces, where \((X_n)_n\) is an increasing fundamental sequence of compact subsets of \(X\) and \(X_n'\) is the locally compact ultraregular space \(X_{n+1} \setminus X_n\) for \(n \in \mathbb{N}\). A key point for the proof of this theorem is the result of van der Put, included at the beginning of this review. When \(X\) is locally compact the theorem is an easy consequence of that result. However, as the author shows in the paper, there exist non-locally compact zero-dimensional spaces \(X\) for which \(C_c(X,K)\) is Fréchet. For this last kind of spaces \(X\) the proof of the theorem is much more involved. The author uses several techniques, on the one hand related to spaces of conti…
reviewed by: Cristina Pérez-García
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 5962615
zbMATH Keywords: non-archimedean Fréchet spaces of continuous functions
zbMATH Keywords: Schauder and orthogonal bases in non-archimedean locally convex spaces
MaRDI profile type: Publication
OpenAlex ID: W1969774050
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DOI: 10.1016/J.JMAA.2011.06.053
description: scientific article; zbMATH DE number 5962615