Pointwise function spaces over compacta are not weak Banach spaces

Tomasz Kania; Jerzy Kąkol

paper

Pointwise function spaces over compacta are not weak Banach spaces

arXiv:2608.17549

Abstract

Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h : C_{p} (K) \to E_{w}$ whose inverse is continuous at $h (0)$ . Consequently, $C_{p} (K)$ and $C_{w} (L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$ . This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_{p} ([0, 1])$ and weak Banach spaces.

5 pp