Proposition 2

— Let E be a Banach space, F a locally convex space and u a linear mapping from E into F. Assume that for every closed subset B of F, every compact subset X of E and every measure $\mu$ on X, the intersection $X \cap u^{-1}(B)$ is $\mu$-measurable. Then u is continuous.

First assume that F is the base field. For every compact subset X of E and every measure $\mu$ on X, the restriction of $u$ to X is $\mu$-measurable (INT, IV). Suppose that $u$ is not continuous. Then we can find a sequence of points $(x_n)$ in E such that $\sum_n \| x_n \| < \infty$ and $|u(x_n)| \geq n$ for every integer $n$. Consider the mapping $g : (t_n) \mapsto \sum_n t_n x_n$ from the cube $C = [0, 1]^{\mathbf{N}}$ into E; it is clear that $g$ is continuous. Hence $f = u \circ g$ is measurable for every measure on C (INT, V); in particular for the measure $\mu$ which is the product of Lebesgue measures on the factors of C. Hence there exists a compact subset D of C such that $\mu(D) > \frac{1}{2}$ and such that the restriction of $f$ to D is continuous, hence also bounded. Let M be the upper bound of $|f|$ on D and let $p \in \mathbf{N}$ be such that $p \geq 4M$. Let $s = (s_n)$ and $t = (t_n)$ be two points of D such that $s_n = t_n$ for all $n \neq p$. Then

Since $|f(s) - f(t)| \leq 2M$ and $|u(x_p)| \geq p \leq 4M$, we get

The Lebesgue-Fubini theorem (INT, V, 2nd ed., § 8, No. 3, cor. 2 of prop. 7) implies that $\mu(D) \leq \frac{1}{2}$; this gives a contradiction. Hence $u$ is continuous.

In the general case, for every $v \in F'$, the linear form $v \circ u$ is continuous, by the preceding argument. Let $(x_n)_{n \in \mathbf{N}}$ be a sequence of points of E tending to 0; then the sequence $(u(x_n))_{n \in \mathbf{N}}$ tends to 0 in F, if F is assigned the topology $\sigma(F, F')$; hence this sequence is bounded for $\sigma(F, F')$ and so it is bounded in F (III, p. 27, cor. 3). Since E is bornological (III, p. 12, prop. 2); the linear mapping $u : E \to F$ is continuous. \*

$^1$ The results of this section depend on the book of Integration.

Exercises