Exercise 6, Appendix 0

Let E be a real vector space, Q, H and $H_0$ positive quadratic forms on E. Assume that $H \leq a \cdot H_0$, where $a$ is a positive real number. Prove that $\operatorname{Tr}(Q/H_0) \leq a \cdot \operatorname{Tr}(Q/H)$. (Reduce to the case that E is finite-dimensional by Remark 1 of No. 1, then conclude by Prop. 1, using the existence of a basis of E that is orthogonal for both H and $H_0$.)