Details

Autor(en) / Beteiligte

• Sánchez, Félix Cabello

Titel

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

Ist Teil von

• Proceedings of the Edinburgh Mathematical Society, 2021-08, Vol.64 (3), p.615-619

Ort / Verlag

Cambridge, UK: Cambridge University Press

Erscheinungsjahr

2021

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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.