Corollary

Let $q$ be a closed positive partial form on E and $u$ the positive self-adjoint operator representing $q$. The domain of $q$ is equal to the domain of $(1_E+u)^{1/2}$, and we have

for every $x\in$ dom($q$).

The domain of $q$ is equal to the domain of $u^{1/2}$ (Prop. 13, a)), which coincides with that of $(1_E+u)^{1/2}($cf. Prop. 3 of IV, p. 271). For every $x\in$ dom($u$)$\subset$ dom($u^{1/2}$), one has

Since the domain of $u$ is dense in the Hilbert space $E_q$ (Prop. 14, a)), this formula extends by continuity to every $x\in$ dom($u^{1/2}$).