The topologization of the space of separately continuous functions

Abstract

Here we introduce locally convex topology $\mathcal{T}$ of the layer uniform convergence on the space $S = CC [0,1] ^ 2$ of all separately continuous functions $f: [0,1] ^ 2 \rightarrow \mathbb{R}$, we prove that the space $(S, \mathcal{T})$ is complete and it is not metrizable one, the space $P$ of all polynomials of two variables on $[0,1] ^ 2$ is everywhere dense in $S$, and so, $S$ is separable.