Example 2

Let $I$ be an interval (bounded or not) in $\mathbf{R}$, and let $E$ be the set of regulated functions (FVR, II, p. 4) defined on $I$ with values in $\mathbf{C}$, having compact support. It is clear that $E$ is a vector space over $\mathbf{C}$; let $f$ be the sesquilinear form $(x, y) \mapsto \int_1^1 \overline{x(t)}\ y(t)\ dt$; it is immediate that $f$ is a positive hermitian form on $E$, and hence defines a prehilbertian structure on this space.