Example 1

Every Fréchet space satisfying the first axiom of countability is a polish space, hence a Lusin space. Consequently, so are the spaces $\mathscr{C}(X)$, where X is locally compact and has a countable base (the topology of $\mathscr{C}(X)$ being that of compact convergence, cf. GT, X, § 3, No. 3, corollary and § 1, No. 6, cor. 3); \* the spaces $\mathscr{C}_0^\infty(U)$, where U is an open subset of $\mathbf{R}^n$ (III, p. 9) and $\mathscr{H}(U)$, where U is an open subset of $\mathbf{C}^n$ (III, p. 10).

Prop. 1 shows that the spaces $\mathscr{C}_0^\infty(U)$, where U is an open set in $\mathbf{R}^n$, $\mathscr{G}_s(I)$, where I is a compact interval in $\mathbf{R}$ and $s \geq 1$, and $\mathscr{H}(K)$, where K is a compact subset of $\mathbf{C}^n$ are all Lusin spaces (III, p. 10). \*