Pointwise function spaces over compacta are not weak Banach spaces

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 $h\colon C_p(K)\to E_w$ 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 K\k{a}kol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.