An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"

TL;DR

This paper investigates when spaces of vector-valued continuous functions over the unit interval and the Cantor set are linearly homeomorphic, revealing conditions under which the underlying quasi Banach space must be locally convex.

Contribution

It extends Kalton's result by providing examples of non-locally convex quasi Banach spaces where the function spaces are isomorphic, showing the original theorem's sharpness.

Findings

Kalton's theorem is sharp; non-locally convex spaces can have isomorphic function spaces.

Constructs specific non-locally convex quasi Banach spaces with a basis where the function spaces are isomorphic.

In metric compacta of finite covering dimension, the spaces are isomorphic to their own continuous functions.

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. It is shown 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 $C(\Delta, X)$ if $K$ is a metric compactum of finite covering dimension.