Exercise 2, Appendix 0

Let E be a real Hilbert space, Q a continuous positive quadratic form on E, and H the quadratic form $x \mapsto \|x\|^2$ on E.

a) Show that $\operatorname{Tr}(Q/H) = \sum_{i \in I} Q(e_i)$ for every orthonormal basis $(e_i)_{i \in I}$ of E. (Let $(a_1, \ldots, a_p)$ be a finite orthonormal family in E; for every $\varepsilon > 0$, there exist a finite subset J of I and elements $a'_1, \ldots, a'_p$ that are linear combinations of the $e_i$ for $i \in J$ and are such that $\|a_j - a'_j\| < \varepsilon$ for $1 \leq j \leq p$; then $\sum_{j=1}^p Q(a'_j) \leq \sum_{i \in J} Q(e_j) \leq \sum_{i \in I} Q(e_i)$. Deduce from this that $\sum_{j=1}^p Q(a_j) \leq \sum_{i \in I} Q(e_i)$.)

b) Deduce from a) a new proof of Prop. 3.

c) Let $E_0$ be a dense linear subspace of E, $Q_0$ (resp. $H_0$) the restriction of Q (resp. H) to $E_0$. Show that $\operatorname{Tr}(Q/H) = \operatorname{Tr}(Q_0/H_0)$. (Let $(e_1, \ldots, e_n)$ be a linearly independent sequence in E, generating a subspace F. Denote by $Q_F$ (resp. $H_F$) the restriction of Q (resp. H) to F. Using Remark 3 of No. 1, show that $\operatorname{Tr}(Q_F/H_F)$ is a continuous function of $(e_1, \ldots, e_n)$.)