On approximation of mappings with values in the space of continuous functions

Abstract

Using a theorem on the approximation of the identity in the Banach space $C_u(Y)$ of all continuous functions $g: Y\rightarrow \mathbb{R}$, defined on a metrizable compact $Y$ with the uniform norm, we prove that for a topological space $X$, a metrizable compact $Y$, a linear subspace $L$ of $Y$ dense in $C_u(Y)$ and a separately continuous function $f: X\times Y\rightarrow \mathbb{R}$ there exists a sequence of jointly continuous functions $f_n: X\times Y\rightarrow \mathbb{R}$ such that $f_n^x = f(x, \cdot)\in L$ and $f_n^x \rightarrow f^x$ in $C_u(Y)$ for each $x\in X$.