Example

Let E be the space of all complex valued continuous functions on $\mathbf{R}$, such that $f(x + n) = f(x)$ for $x \in \mathbf{R}$ and $n \in \mathbf{Z}$. We assign to E the scalar product defined by

Then E is a Hausdorff prehilbertian space, but is not complete. For every integer $n \in \mathbf{Z}$, let $e_n(x) \in \mathbf{e}(nx)$. It is immediate that the family $(e_n)_{n \in \mathbf{Z}}$ is orthonormal in E. Moreover, the topology of uniform convergence on E is finer than the topology deduced from the norm $\|f\|_2 = \langle f|f \rangle^{1/2}$. The family $(e_n)_{n \in \mathbf{Z}}$ is total in E for the uniform convergence (GT, X, § 4, No. 4), and a fortiori in the prehilbertian space E. Hence $(e_n)_{n \in \mathbf{Z}}$ is an orthonormal basis of E.