Corollary

— The hypotheses on E being as in th. 2, assume, in addition that every bounded subset of E is relatively compact. Then the strong dual of E is a Lusin space. In particular, the strong dual of a Fréchet space satisfying the first axiom of countability, which is also a Montel space, is a Lusin space.

\* Example 2. — Let U be an open subset of $\mathbf{R}^n$. The corollary applies in particular to the Fréchet space $E = \mathscr{C}_c^\infty(U)$; its dual $\mathscr{C}_c^{-\infty}(U)$ (the space of distributions with compact support on U) is then a Lusin space.

The space $\mathscr{C}_c^\infty(U)$ is a strict inductive limit of a sequence of Fréchet spaces $\mathscr{C}_{K_n}^\infty(U)$ satisfying the first axiom of countability (III, p. 9). We can show that each of the spaces $\mathscr{C}_{K_n}^\infty(U)$ is a Montel space; in addition, every bounded subset of $\mathscr{C}_c^\infty(U)$ is contained in one of the spaces $\mathscr{C}_{K_n}^\infty(U)$ (III, p. 5, prop. 6). We can then apply the corollary of th. 2. Then the dual $\mathscr{C}_c^{-\infty}(U)$ of $\mathscr{C}_c^\infty(U)$ (the space of distributions on U) is a Lusin space for the strong topology.

Similarly we prove that for every open subset U of $\mathbf{C}^n$, and for every compact subset K of $\mathbf{C}^n$, the strong dual of $\mathscr{H}(U)$ and the strong dual of $\mathscr{H}(K)$ are Lusin spaces. \*