Isomorphisms of spaces of norm-continuous functions
type: Item
instance of: scholarly article
title: Isomorphisms of spaces of norm-continuous functions
zbMATH Open document ID: 0569.46018
author: Michael Cambern
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: If X is a compact Hausdorff space and E a Banach space, then C(X,E) denotes the space of continuous functions on X to E provided with the supremum norm. If E is the one-dimensional field of scalars then we write C(X) for C(X,E). The well known Banach-Stone theorem states that, for compact X and Y, if C(X) and C(Y) are isometric then X and Y are homeomorphic. Various authors, beginning with M. Jerison, have considered the problem of determining geometric properties of E which allow generalizations of this theorem to spaces of norm-continuous vector functions C(X,E). Another type of generalization of the theorem, while still dealing with scalar functions, replaces isometries by isomorphisms T with \(\| T\| \| T^{-1}\|\) small. In this article these two directions of generalization are combined. It is shown that if E is uniformly convex then the existence of an isomorpism T of C(X,E) onto C(Y,E), with \(\| T\| \| T^{- 1}\|\) less than a constant depending on the modulus of convexity associated with E, implies that X and Y are homeomorphic.
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 3909096
zbMATH Keywords: compact Hausdorff space
zbMATH Keywords: Banach-Stone theorem
zbMATH Keywords: uniformly convex
zbMATH Keywords: modulus of convexity
MaRDI profile type: Publication
full work available at URL: pjm.1985.116.243
OpenAlex ID: W2046069765
DOI: 10.2140/PJM.1985.116.243
description: scientific article; zbMATH DE number 3909096