On approximation of the separately and jointly continuous functions

Abstract

We investigate the following problem: which dense subspaces $L$ of the Banach space $C(Y)$ of continuous functions on a compact $Y$ and topological spaces $X$ have such property, that for every separately or jointly continuous functions $f: X\times Y \rightarrow \mathbb{R}$ there exists a sequence of separately or jointly continuous functions $f_{n}: X\times Y \rightarrow \mathbb{R}$ such, that $f_n^x=f_n(x, \cdot) \in L$ for arbitrary $n\in \mathbb{N}$, $x\in X$ and $f_n^x\rightrightarrows f^x$ on $Y$ for every $x\in X$? In particular, it was shown, if the space $C(Y)$ has a basis that every jointly continuous function $f: X\times Y \rightarrow \mathbb{R}$ has jointly continuous approximations $f_n$ such type.