Corollary

Let X be a compact complex analytic manifold and E an analytic vector bundle over X, locally of finite rank. The vector space $\mathscr{S}^{\omega}(X; E)$ is finite-dimensional.

Since it is compact, the analytic manifold X is locally finite-dimensional. By Proposition 11, the identity mapping of $\mathscr{S}^{\omega}(X; E)$ is a compact linear mapping. This implies that the space $\mathscr{S}^{\omega}(X; E)$ is finite-dimensional (Remark 3 of III, p. 2).