Fréchet spaces of non-archimedean valued continuous functions
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 continuous functions, and on the other hand related to the theory of orthogonal bases in non-archimedean locally convex spaces. As it is usual when one sees a rather hard proof, like the one given in this paper for the above theorem, one question crosses one's mind: Is it possible to give a shorter or easier proof of that theorem? An interesting but probably difficult task!
non-archimedean Fréchet spaces of continuous functions
Schauder and orthogonal bases in non-archimedean locally convex spaces
scientific article; zbMATH DE number 5962615
scientific article