Exercise 11, § 3

Let $T$ be a metrizable topological space, $E$ a topological vector space which is a Baire space, and $M$ a family of mappings from $E \times T$ into a topological vector space $F$, satisfying the following conditions : 1° for all $t_0 \in T$, the set of all mappings $x \mapsto f(x, t_0)$ where $f$ runs through $M$, is an equicontinuous set of linear mappings from $E$ into $F$; 2° for all $x_0 \in E$, the set of all mappings $t \mapsto f(x_0, t)$ from $T$ into $F$, where $f$ runs through $M$, is equicontinuous. Show that $M$ is equicontinuous. (Given $t_0 \in T$ and a closed balanced neighbourhood $V$ of 0 in F, for every $x \in E$, let $d_x$ be the upper bound of the radii of all open balls with center $t_0$ in T such that, for an arbitrary point $t$ in one of these balls, we have $f(x, t) - f(x, t_0) \in V$ for all $f \in M$. Show that $x \mapsto d_x$ is upper semi-continuous at every point $x_0 \in E$; for this, show that if we had $d_x > \alpha > d_{x_0}$ for points arbitrarily close to $x_0$, then for every neighbourhood W of 0 in F, $f(x_0, t) - f(x_0, t_0)$ would belong to $V + W$ for $d(t, t_0) \leq \alpha$ and $f \in M$. Finally use GT, IX, § 5, No. 4, th. 2.)