  $C_p(X)$ 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 $\ell_{1}$.
