Remark

— 1) For every subspace F of E, let $Q_F$ denote the restriction of Q to F and $H_F$ that of H. We have $\operatorname{Tr}(Q_F/H_F) \leq \operatorname{Tr}(Q/H)$ and $\operatorname{Tr}(Q/H)$ is the upper bound of the set of all numbers $\operatorname{Tr}(Q_F/H_F)$ where $F$ ranges over the family of all finite dimensional vector subspaces of $E$.

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 $\operatorname{Tr}(Q/H) = \operatorname{Tr}(Q_1/H_1)$.