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

2021/02/03 by Félix Cabello Sánchez, Sánchez, Félix Cabello
Mathematics · #46A16 #46E10 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2102.02102

openalex publication_date 2021/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The paper alluded to in the title contains the following striking result: Let\nI be the unit interval and \Δ the Cantor set. If X is a quasi Banach\nspace containing no copy of c₀ which is isomorphic to a closed subspace of a\nspace with a basis and C(I, X) is linearly homeomorphic to C(\Δ, X),\nthen X is locally convex, i.e., a Banach space.\n It is shown that Kalton result is sharp by exhibiting non locally convex\nquasi Banach spaces X with a basis for which C(I, X) and C(\Δ, X) are\nisomorphic. Our examples are rather specific and actually in all cases X is\nisomorphic to C(\Δ, X) if K is a metric compactum of finite covering\ndimension.\n

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