
arXiv: 2102.02102
AbstractThe 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.
spaces of vector-valued continuous functions, Mathematics - Functional Analysis, Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.), 46A16, 46E10, General Topology (math.GN), FOS: Mathematics, Topological linear spaces of continuous, differentiable or analytic functions, quasi Banach spaces, Mathematics - General Topology, Functional Analysis (math.FA)
spaces of vector-valued continuous functions, Mathematics - Functional Analysis, Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.), 46A16, 46E10, General Topology (math.GN), FOS: Mathematics, Topological linear spaces of continuous, differentiable or analytic functions, quasi Banach spaces, Mathematics - General Topology, Functional Analysis (math.FA)
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