Example

Let I be the interval $(-1, 1)$ of $\mathbf{R}$ and E the vector space of real valued continuous functions on I. Let $x$ denote the canonical injection from I into $\mathbf{R}$, considered as an element of E. By the Stone-Weierstrass theorem the sequence $(x^n)_{n \in \mathbf{N}}$ is total in E for the topology of uniform convergence GT, X, § 4, No. 2). Consider E as a real Hausdorff prehilbertian space in which the scalar product is given by

The sequence $(x^n)_{n \in \mathbf{N}}$ is then total in the prehilbertian space E. Let $(\Pi_n)_{n \in \mathbf{N}}$ be the sequence obtained by the orthonormalisation of the sequence $(x^n)_{n \in \mathbf{N}}$. We can show that $\Pi_n = (n + \frac{1}{2})^{1/2} P_n$, where the Legendre polynomial $P_n$ is defined by