On the weak and pointwise topologies in function spaces

For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that $K$ is an infinite (metrizable) compact space. Is it true that $C_w(K)$ and $C_p(K)$ are homeomorphic? We show that the answer is “no”, provided $K$ is an infinite compact metrizable $C$-space. In particular our proof works for any infinite compact metrizable finite-dimemsional space $K$.