Exercise 5, Appendix 0

Let E be a real vector space, $(\lambda_i)_{i \in I}$ a summable family of positive numbers, and $(y_i)_{i \in I}$ a family of linear forms on E. For every $x \in E$, set $H(x) = \sum_{i \in I} \langle x, y_i \rangle^2$ and $Q(x) = \sum_{i \in I} \lambda_i \langle x, y_i \rangle^2$; assume that $H(x)$ is finite for every $x \in E$. Show that $Q(x)$ is finite for every $x \in E$, that Q and H are positive quadratic forms on E, and that $\operatorname{Tr}(Q/H) \leq \sum_{i \in I} \lambda_i$. (Set $u(x) = (\langle x, y_i \rangle)_{i \in I}$ and apply Exer. 3 to the linear mapping $u$ of E into $l_2(I)$.)