Proposition 7

Let $u$ and $v$ be positive compact endomorphisms of E.

a) One has $|\lambda_n(u)-\lambda_n(v)|\leqslant \|u-v\|$ for every $n\in I$;

b) If $u\leqslant v$, then $\lambda_n(u)\leqslant \lambda_n(v)$ for every $n\in I$.

Let $n\in I$ and let F be a vector subspace of dimension $n$ of E. For every $x\in F$, one has

hence the inequalities

Assertion a) follows from this by Proposition 6 of IV, p. 154.

If $u\leqslant v$, one has $\langle x|u(x)\rangle \leqslant \langle x|v(x)\rangle$ for every $x\in E$. For every $n\in I$ and every subspace F of dimension $n$, one therefore has $R_F(u)\leqslant R_F(v)$, whence $\lambda_n(u)\leqslant \lambda_n(v) ($loc. cit.).