Proposition 4

— Let E be a Hausdorff prehilbertian space, $\hat{E}$ the normed space completion of E(GT, IX, § 3, No. 3). The scalar product* $(x, y) \mapsto \langle x|y \rangle$ extends by continuity to a positive and separating hermitian form on* $\hat{E}$, and defines a hilbertian space structure on $\hat{E}$.

The existence of the extension of $(x, y) \mapsto \langle x|y \rangle$ to $\hat{E} \times \hat{E}$ follows from the continuity of this sesquilinear form an $E \times E$ (GT, III, § 6, No. 5, th. 1). Moreover, this extension, which will also be denoted by $(x, y) \mapsto \langle x|y \rangle$ is a hermitian form and satisfies the relation $\langle x|x \rangle = \|x\|^2$, by virtue of the principle of extension of identities ($\|x\|$ being the norm on $\hat{E}$ obtained by extending the norm on E by continuity) ; this proves that the relation $\langle x|x \rangle = 0$ implies $x = 0$ in $\hat{E}$, hence that the form $(x, y) \mapsto \langle x|y \rangle$ is positive and separating, and consequently defines a hilbertian space structure on $\hat{E}$. Q.E.D.

This hilbertian space is said to be the completion of the Hausdorff prehilbertian space E.

\* Example 6. — Let U be an open subset of $\mathbf{R}^n$ ($n \geqslant 1$). Let $\mathscr{C}_0^1(\mathrm{U})$ be the vector space of all functions of class $\mathrm{C}^1$ with compact support in U. We define a Hausdorff prehilbertian space structure on $\mathscr{C}_0^1(\mathrm{U})$ whose scalar product is given by

This prehilbertian space is not complete. Its completion is called the Dirichlet space associated with U. \*