Exercise 3, § 1

Soit E a complex Hilbert space and let $B = (e_i)_{i\in I}$ be an orthonormal basis of E. For all $i\in I$, we denote by $p_i$ the orthoprojector of image $\mathbf{C}e_i$.

a) Let K be a precompact subset of E. For every $\varepsilon >0$, there exists a finite subset J of I such that, for every $x\in K,\|x-\sum_{i\in J}p_j(x)\|\leqslant \varepsilon$.

b) Let $u\in \mathscr{D}_B(E)$ and denote by $\lambda = (\lambda_i)_{i\in I}$ its family of eigenvalues. The family $(\lambda_ip_i)$ is summable in $\mathscr{L}(E)$ endowed with the topology of precompact convergence and its sum is equal to $u($cf. prop. 1 of IV, p. 147)