Definition 3

Let F be a Hilbert space and let $u$ be a compact linear mapping of E into F.

For every integer $n\in I_E$ one denotes by $\alpha_n(u) =\lambda_n(u^*\circ u)$. Let J be the set of $n\in I_E$ such that $\alpha_n(u)>0$. The family $(\alpha_n(u))_{n\in J}$ is called the sequence of singular values of $u$ repeated according to multiplicity.

One says that the sequence $(\alpha_n(u))_{n\in I_E}$ is the extended sequence of singular values of $u$.

The sequence $(\alpha_n(u))_{n\in I_E}$ is well defined since the endomorphism $u^*\circ u$ of E is compact (III, p. 5, prop. 3); it is a decreasing family of positive real numbers since $u^*\circ u$ is positive.