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An example regarding Kalton's paper ‘isomorphisms between spaces of vector-valued continuous functions’

Part of: Linear function spaces and their duals Topological linear spaces and related structures

Published online by Cambridge University Press:  02 August 2021

Félix Cabello Sánchez
Félix Cabello Sánchez*
Affiliation:
Departamento de Matemáticas and IMUEx, Universidad de Extremadura, Avenida de Elvas, 06071Badajoz, Spain (fcabello@unex.es)
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Abstract

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which is isomorphic to a closed subspace of a space with a basis and $C(I,\,X)$ is linearly homeomorphic to $C(\Delta ,\, X)$, then $X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces $X$ with a basis for which $C(I,\,X)$ and $C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, $X$ is isomorphic to $C(K,\,X)$ if $K$ is a metric compactum of finite covering dimension.

Keywords

spaces of vector-valued continuous functionsquasi Banach spaces

MSC classification

Primary: 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
Secondary: 46E10: Topological linear spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 615 - 619
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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