A note on Banach spaces $E$ admitting a continuous map from $C_p(X)$ onto $E_{w}$

TL;DR

This paper investigates the conditions under which certain continuous maps from spaces of continuous functions on Tychonoff spaces onto Banach spaces with weak topology exist, revealing restrictions on the Banach spaces and the underlying spaces involved.

Contribution

It proves that linear, sequentially continuous surjections imply the Banach space is finite-dimensional, and characterizes spaces homeomorphic to $C_p(X)$, showing they must have specific topological and Banach space properties.

Findings

Linear, sequentially continuous maps imply finite-dimensionality.

Homeomorphisms require the underlying space to be a countable union of non-scattered compact sets.

Banach spaces in such homeomorphisms must contain an isomorphic copy of $oldsymbol{ extit{ extltilde}}_{1}$.

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 $\ell_{1}$.