Proposition 2

— Let f be a continuous linear form defined over a vector subspace M, of a locally convex space E; then there exists a continuous linear form h that is defined over E and is an extension of f.

From the corollary above and II, p. 7, cor. 2, there exists a continuous semi-norm $p$ on E, such that $|f(y)| \leq p(y)$ for all $y \in M$. By the Hahn-Banach th. (II, p. 23, cor. 1) there exists a linear form $h$ on E that extends $f$ and is such that $|h(x)| \leq p(x)$ for all $x \in E$, and this implies that $h$ is continuous (II, p. 6, prop. 5).