Remark

Let $c\in \mathbf{R}_+$. Let $q$ be a closed partial hermitian form with dense domain such that $q(x, x)\geqslant -c\|x\|^2$ for every $x$ belonging to the domain of $q$. This means that the closed partial hermitian form with domain dom($q$) defined by $\widetilde{q}(x, y) =q(x, y) +c\langle x|y\rangle$ is a positive partial form. By Theorem 3, such forms therefore correspond to the self-adjoint partial operators $u$ on E such that $u+c1_E$ is positive, that is to say such that $u$ is bounded below (Def. 7 of IV, p. 237) by $-c$.

Conversely, let $u$ be a symmetric partial operator on E such that

for every $x\in$ dom($u$). Then $v=u+c1_E$ is a positive partial operator on E. The Friedrichs extension of $u$ is defined to be the self-adjoint operator $\widetilde{v}-c1_E$, where $\widetilde{v}$ is the Friedrichs extension of $v$; it is a self-adjoint extension of $u$, which does not depend on the choice of the real number $c$ satisfying (17).