English

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

Functional Analysis 2021-05-28 v2 General Topology

Abstract

The paper alluded to in the title contains the following striking result: Let II be the unit interval and Δ\Delta the Cantor set. If XX is a quasi Banach space containing no copy of c0c_0 which is isomorphic to a closed subspace of a space with a basis and C(I,X)C(I, X) is linearly homeomorphic to C(Δ,X)C(\Delta, X), then XX 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)C(I, X) and C(Δ,X)C(\Delta, X) are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to C(Δ,X)C(\Delta, X) if KK is a metric compactum of finite covering dimension.

Keywords

Cite

@article{arxiv.2102.02102,
  title  = {An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"},
  author = {Félix Cabello Sánchez},
  journal= {arXiv preprint arXiv:2102.02102},
  year   = {2021}
}

Comments

4 pages

R2 v1 2026-06-23T22:48:12.716Z