Example 4

Let $\ell^2$ (or $\ell^2(\mathbf{N})$) be the set of sequences $x = (x_n)_{n \in \mathbf{N}}$ of elements of $K$ such that $\sum_{n=0}^\infty |x_n|^2$ is finite. One can show that $\ell^2$ is a vector subspace of $K^\mathbf{N}$ and define a prehilbertian space structure on $\ell^2$ by means of the hermitian form $(x, y) \mapsto \sum_{n=0}^\infty \overline{x_n} y_n$ (cf. V, p. 18).