Corollary 3

Let T, X be topological spaces, let Y be a uniform space and let f be a mapping of T × X into Y. Then f is continuous, provided that the following conditions are satisfied:

1) For each $x \in X$, the partial mapping $t \to f(t, x)$ is continuous.

2) As t runs through T, the partial mappings $x \to f(t, x)$ form an equicontinuous subset of $\mathscr{F}(X; Y)$.

In particular, take T to be a subset H of $\mathscr{F}(X; Y)$ and f to be the canonical mapping $(h, x) \to h(x)$ of $H \times X$ into Y; condition 1) of Corollary 3 means that H is endowed with a topology finer than that of pointwise convergence, and condition 2) means that H is equicontinuous. Hence: