Corollary 4

Let E be a locally convex space. If every continuous seminorm on E is majorized by a continuous prehilbertian seminorm, then E is an approximation space.

Let $\mathscr{P}$ be the set of continuous prehilbertian seminorms on E. For each $p\in \mathscr{P}$, the hypothesis implies that the seminormed space $E_p$ obtained by endowing E with the seminorm $p$ is an approximation space (Corollary 2 and Proposition 10), and the identity mapping of E into $E_p$ is continuous. The corollary therefore follows from Proposition 12.