Definition 4

— An orthonormal and total family in a Hausdorff prehilbertian space E is called an orthonormal basis of E.

An orthonormal basis of a Hausdorff prehilbertian space E is also an orthonormal basis of the completion of E.

Let $(e_i)_{i \in I}$ be an orthonormal basis of E; for every $x \in E$, the numbers $\langle e_i | x \rangle$ are called, by abuse of language, the coordinates of $x$ with respect to the basis $(e_i)$. For every $x$ and $y$ in E, we have

An orthonormal basis of E is not, in general, a basis of E over the field K in the sense defined in A, II, p. 25; to avoid any confusion we shall always say that a basis of a prehilbertian space E in the sense of loc. cit. is an algebraic basis of E over K.

Let E and F be two Hausdorff prehilbertian spaces and $u$ a continuous linear mapping from E into F. Let $(e_i)_{i \in I}$ (resp. $(f_j)_{j \in J}$) be an orthonormal basis of E (resp. F). Put

for $i \in I,\ j \in J$. The family $(u_{ji})_{(i,j) \in I \times J}$ is called the matrix of $u$ with respect to the orthonormal bases $(e_i)$ and $(f_j)$. Let $x \in E$ and $y = u(x)$; if we write $\xi_i = \langle e_i | x \rangle$ and $\eta_j = \langle f_j | y \rangle$ for the coordinates of $x$ and $y$ respectively, we get $\eta_j = \sum_{i \in I} u_{ji} \xi_i$ for all $j \in J$. When $(e_i)$ is an algebraic basis of E and $(f_j)$ an algebraic basis of F, our definition is consistent with that of A, II, § 10, No. 4.