Corollary 1

— Let $v$ be a continuous, positive endomorphism with finite trace of the hilbertian space $E$. There exists an orthonormal basis $(e_i)_{i \in I}$ of $E$ and a summable family of positive real numbers $(\lambda_i)_{i \in I}$ such that $v(e_i) = \lambda_i e_i$ for all $i \in I$.

Let $\Phi(x, y) = \langle x|v(y)\rangle$ for $x, y$ in $E$. Then $\Phi$ is a positive hermitian form on $E$. There exists (V, p. 8, corollary) a hilbertian space $F$ and a continuous linear mapping $u$ from $E$ into $F$ such that $\Phi(x, y) = \langle u(x)|u(y)\rangle$ for $x, y$ in $E$. In other words, we have $v = u^*u$. By virtue of def. 9 (V, p. 52), $u$ is a Hilbert-Schmidt mapping from $E$ into $F$. By th. 2, there exists an orthonormal basis $(e_i)_{i \in I}$ of $E$ such that the vectors $u(e_i)$ are two by two orthogonal. Let $i \in I$; for every $j \neq i$ in $I$, we have

hence $v(e_i)$ is proportional to $e_i$ and is of the form $\lambda_i e_i$, where $\lambda_i = \langle e_i | v(e_i) \rangle$; then