On approximation of mappings with values in the space of continuous functions - Voloshyn, H.A.; Maslyuchenko, V.K.; Nesterenko, O.N. (Carpathian Mathematical Publications, 2012)
Using a theorem on the approximation of the identity in the Banach space $C_u(Y)$ of all continuous functions $g: Y→ \mathbb{R}$, 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 $f: X× Y→ \mathbb{R}$ there exists a sequence of jointly continuous functions $f_n: X× Y→ \mathbb{R}$ such that $f_n^x = f(x, ·)\in L$ and $f_n^x → f^x$ in $C_u(Y)$ for each $x\in X$.