Lemma 5

Let $X_0$ be the underlying real analytic manifold of X and $E_0$ the complex vector bundle on $X_0$ underlying E.

a) The vector subspace $\mathscr{S}^{\omega}(X; E)$ of $\mathscr{S}^0(X_0; E_0)$ is closed, and the injection of $\mathscr{S}^{\omega}(X; E)$ into $\mathscr{S}^0(X_0; E_0)$ is continuous and strict ;

b) The canonical injection of $\mathscr{S}^{\omega}(X; E)$ into $\mathscr{S}^1(X_0; E_0)$ is continuous.

Suppose first that there exist an integer $n\geqslant 0$ and a Banach space F such that X is an open subset of $\mathbf{C}^n$ and E the trivial vector bundle $X\times F$. In this case, the topological vector spaces $\mathscr{S}^{\omega}(X; E)$, $\mathscr{S}^0(X_0; E_0)$ and $\mathscr{S}^1(X_0; E_0)$ are isomorphic to $\mathscr{C}^{\omega}(X; F),\mathscr{C}^0(X_0; F)$ and $\mathscr{C}^1(X_0; F)$ respectively, and the vector space $\mathscr{S}^0(X_0; E_0)$ is metrizable (TG, X, p. 20, cor. of prop. 1). The lemma then follows from VAR, 3.3.2, p. 28.

Let us pass to the general case. Let $\mathfrak{F}$ be a filter on $\mathscr{S}^{\omega}(X; E)$ which converges in the space $\mathscr{S}^0(X_0; E_0)$ to a limit $f$. It is a question of proving that $f$ belongs to $\mathscr{S}^{\omega}(X; E)$ and that the filter $\mathfrak{F}$ converges to $f$ in the space $\mathscr{S}^1(X_0; E_0)$. This statement is of a local nature, and therefore follows from the first part of the proof.