In their book, Reed & Simon state the theorem (Theorem V.3):
Let $X$ be a locally convex space and let $Y \subset X$ be a subspace. Let $\ell: Y \rightarrow \mathbb{R}$ (or $\mathbb{C}$ if $X$ is a complex space) be linear and continuous. Then, there is a continuous linear map $L: X \rightarrow \mathbb{R}$ (or $\mathbb{C}$) with $L|_Y = \ell$.
Their proof is:
The relative topology on $Y$ is given by the restrictions of the continuous seminorms to $Y$. Thus, $|\ell(x)| \leq C\rho(x)$ for some continuous seminorm. Applying the Hahn-Banach extension theorem, we obtain our result.
- How do they conclude that $|\ell(x)| \leq C\rho(x)$ for some continuous seminorm? More precisely, how does this follow from $Y$ being the restriction of the continuous seminorms that give the topology on $X$?
- Following their proof, they note "thus, locally convex spaces possess many continuous linear functionals; in fact, enough to separate points". How does this remark follow from the theorem? Intuitively, I do not see the connection.