Corollary 2

Let F be a Hilbert space and let $u$ be a compact continuous linear mapping from E into F. There exists a countable set I, an orthonormal basis $(e_i)_{i\in I}$ of the initial space Ker($u$)$^{\circ}$ of $u$, an orthonormal family $(f_i)_{i\in I}$ of F and a family $(\alpha_i)_{i\in I}\in (\mathbf{R}^*_+)^I$ such that $u(e_i) =\alpha_if_i$ for all $i\in I$.

Let $v=u^*\circ u$. It is a compact endomorphism (III, p. 5, Prop. 3) and positive, hence hermitian, of E. By Corollary 1, there exists an orthonormal basis $(e_j)_{j\in J}$ of E such that $v$ is diagonal in this basis. The family $(\lambda_j)_{j\in J}$ of its eigenvalues is contained in $\mathbf{R}_+^J$. Put $\alpha_j=\surd\overline{\lambda_j}$ for all $j\in J$. Let I be the set of $j\in J$ such that $\alpha_j\not = 0$. It is a countable set since $v$ is compact. The family $(e_i)_{i\in I}$ is an orthonormal basis of the initial space of $v$, which is the initial space Ker($u$)$^{\circ}$ of $u$ (EVT, V, p. 43, Prop. 8). Put $f_i=\frac{1}{\alpha_i}u(e_i)$ for $i\in I$. Whatever $i$ and $j$ in I may be, one has

whence it follows that the family $(f_i)_{i\in I}$ is orthonormal in F. The corollary follows, since $u(e_i) =\alpha_if_i$ for all $i\in I$.