Proposition 11 (Weyl Inequalities)

Let G be a Hilbert space and $v\in \mathscr{L}^c(F; G)$. Let $g:\mathbf{R}_+\rightarrow [-\infty ,+\infty [$ be an increasing function such that the function $g\circ$ exp is convex. We have

for all $n\in I_E\cap I_F$.

Put $I = [-\infty ,+\infty [$ and $f=g\circ$ exp. We may apply Lemma 5 with

$a_i=$ log($\alpha_i(v\circ u)$)$\in I,b_i=$ log($\alpha_i(v)\alpha_i(u)$)$\in I$

and $\varrho_i= 1$ for $0\leqslant i\leqslant n$, since

$\sum_{i=0}^ja_i=$ log$(\prod_{i=0}^j\alpha_i(v\circ u))\leqslant$ log$(\prod_{i=0}^j\alpha_i(v)\alpha_i(u))=\sum_{i=0}^jb_i$

for $0\leqslant j\leqslant n$ by the corollary to Prop. 10. Inequality (6) is then the desired conclusion.