Theorem 2

Let E and F be two hilbertian spaces and u a Hilbert-Schmidt mapping from E into F. There exists an orthonormal basis $(e_i)_{i \in I}$ of E which is transformed by u into an orthogonal family in F.

Let B denote the (closed) unit ball of E, with the weakened topology assigned to it; this is a compact space (V, p. 17). We put $Q(x) = \|u(x)\|^2$ for all $x \in B$. Finally let P denote the set of all vectors x in E satisfying the following property ;

(H) For every $y \in E$ orthogonal to x, the element $u(y)$ of E is orthogonal to $u(x)$.