Exercise 4, Appendix 0

Let I be a set and $l^2(I)$ the vector space of families $x = (x_i)_{i \in I}$ of real numbers such that $\sum_{i \in I} x_i^2$ is finite. Let $(\lambda_i)_{i \in I}$ be a summable family of positive numbers. For every $x$ in $l^2(I)$, set $Q(x) = \sum_{i \in I} \lambda_i x_i^2$ and $H(x) = \sum_{i \in I} x_i^2$. Show that $\operatorname{Tr}(Q/H) = \sum_{i \in I} \lambda_i$. (Make use of Exercise 2 a).)