Proposition 9

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

a) For $n\in I_E$, one has

$\alpha_n(u) =$ sup$_{F\in\mathscr{F}_{n+1}}$ inf$_{x\in F-\{0\}}\frac{\|u(x)\|}{\|x\|}=$ inf$_{F\in\mathscr{F}_n}$ sup$_{x\in F^{\circ-}\{0\}}\frac{\|u(x)\|}{\|x\|}$.

b) Let J be the set of $n\in I_E$ such that $\alpha_n(u)\not = 0$. There exist orthonormal families $(e_n)_{n\in J}$ in E and $(f_n)_{n\in J}$ in F such that for every $x\in E$, one has

Since $\langle x|u^*u(x)\rangle =\|u(x)\|^2$ for every $x\in E$, the definitions and prop. 6 of IV, p. 154, a), imply the equality of the first assertion.

Let $(e_n)_{n\in I_E}$ be an orthonormal family of E satisfying the conclusions of prop. 3 of IV, p. 152 applied to the positive compact endomorphism $u^*\circ u$ of E. Put $f_n=\alpha^{-1}_nu(e_n)$ for $n\in J$. By reasoning as in the proof of corollary 2 of IV, p. 150, one obtains assertion c).