Remark

Let $q$ be a closed positive partial form and $q'$ a continuous positive Hermitian form on E. The positive Hermitian form defined on dom($q$) by $(x, y)\mapsto q(x, y) +q'(x, y)$ is a closed positive partial Hermitian form with the same domain as $q$. It is denoted by $q+q'$.

By EVT, V, p. 16, Cor. 2, there exists a unique linear mapping $u'\in \mathscr{L}(E)$ such that $q'(x, y) =\langle x|u'(y)\rangle$ for every $(x, y)\in E\times E$. The endomorphism $u'$ is positive and the partial operator representing the closed positive partial Hermitian form $q+q'$ is $u+u'$.