Remark

— 1) For every subspace $F$ of $E$, denote by $Q_F$ the restriction of $Q$ to $F$, and by $H_F$ that of $H$. Then $\operatorname{Tr}(Q_F/H_F) \leq \operatorname{Tr}(Q/H)$ and $\operatorname{Tr}(Q/H)$ is the supremum of the numbers $\operatorname{Tr}(Q_F/H_F)$ for $F \subset E$ finite-dimensional.

2) Let $E_1$ be a real vector space, $Q_1$ and $H_1$ two positive quadratic forms on $E_1$, and $\pi : E \to E_1$ a surjective linear mapping. If $Q = Q_1 \circ \pi$ and $H = H_1 \circ \pi$, then $\operatorname{Tr}(Q/H) = \operatorname{Tr}(Q_1/H_1)$.

(1) Cf. TVS, V, §4, No. 9.