Remark 3

Let $u$ be a positive compact endomorphism of E. Let $B = (f_i)_{i\in I}$ be an orthonormal basis of E such that $u$ is diagonal in the basis B (Theorem 1), and let $(\lambda_i)_{i\in I}$ be the family of eigenvalues of $u$ in this basis. Let J be the set of $i\in I$ such that $\lambda_i>0$; the family $(e_i)_{i\in J}$ is an orthonormal basis of the space Ker($u$)$^{\circ}$. For all $i\in J$, put $e_i=f_i$ and $\alpha_i=\lambda_i$. It follows that $u(e_i) =\alpha_if_i:$ the family $(\alpha_i)_{i\in J}$ is the family of singular values of $u$ relative to the basis $(e_i)_{i\in J}$.