A note on Banach spaces $E$ admitting a continuous map from $C_p(X)$ onto $E_w$

Abstract

$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the topology of pointwise convergence. A Banach space $E$ equipped with the weak topology is denoted by $E_w$. It is unknown whether $C_p(K)$ and $C(L{)}_w$ can be homeomorphic for infinite compact spaces $K$ and $L$ \cite{Krupski-1}, \cite{Krupski-2}. In this paper we deal with a more general question: what are the Banach spaces $E$ which admit certain continuous surjective mappings $T:C_p(X)\to E_w$ for an infinite Tychonoff space $X$? First, we prove that if $T$ is linear and sequentially continuous, then the Banach space $E$ must be finite-dimensional, thereby resolving an open problem posed in \cite{Kakol-Leiderman}. Second, we show that if there exists a homeomorphism $T:C_p(X)\to E_w$ for some infinite Tychonoff space $X$ and a Banach space $E$, then (a) $X$ is a countable union of compact sets $X_n,n\in \omega$, where at least one component $X_n$ is non-scattered; (b) $E$ necessarily contains an isomorphic copy of the Banach space ${\ell }_1$.

13 pages