$L^2$ and Schwartz Space

Question

It is stated, Introduction to the theory of Pseudodiffernetial Operators, by Raymond, pg 9, Theorem 16, if $S$ is the Schwartz space

If $\varphi \mapsto U(\varphi)$ is a semilinear form on $ S$ satisfying $|U(\varphi)|\le C||\varphi||_2$, then there exists a unique $u \in L^2$ such that $U(\varphi) = (u, \varphi)$ for $\varphi\in S$ and one has $||u||_2 \le C$.

I don't see how this is true from their argument:

They said we use Riesz Representation Theorem - but $S \subsetneq L^2$. So is this statement true? If so, how do we prove it?

Answer 1

Extend $U$ by Hahn Banach Theorem and then apply Riesz' Theorem.