A note on Banach spaces E admitting a continuous map from Cp(X) onto Eω

Cp(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 Ew. It is unknown whether Cp(K) and C(L)ω 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:Cp(X)→Eω 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:Cp(X)→Ew for some infinite Tychonoff space X and a Banach space E, then (a) X is a countable union of compact sets Xn,n∈ω, where at least one component Xn is non-scattered; (b) E necessarily contains an isomorphic copy of the Banach space ℓ1.