On approximation of the separately and jointly continuous functions

H. A. Voloshyn, V. K. Maslyuchenko, O. V. Maslyuchenko

Abstract


We investigate the following problem: which dense subspaces$L$ of the Banach space $C(Y)$ of continuous functions on acompact $Y$ and topological spaces $X$ have such property, thatfor every separately or jointly continuous functions $f: X\times Y\rightarrow \mathbb{R}$ there exists a sequence of separately orjointly 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 $f_n^x\rightarrow f^x$ uniformly 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.

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