The space of all real continuous function on a Tichonoff space $X$ equipped with the pointwise convergence topology is denoted by $C_p(X)$. By $E_w$ we denote an infinite dimensional real Banach space $E$ equipped with the weak topology.
The lecture is devoted to the question whether there exist a Banach space $E$ and a Tichonoff space $X$ such that spaces $C_p(X)$ and $E_w$ are homeomorphic. We show that if such homeomorphism exists, then (1) $X$ is a union of a sequence of compact sets where at least one term is non-scattered and (2) $E$ contains a subspace isomorphic to $l_1$.

The talk is based on the joint work with J. Kąkol and A. Leiderman.