Theorem 3

The mapping which associates with a positive self-adjoint operator $u$ on E the positive partial form $q_u$ is a bijection between the set of positive self-adjoint partial operators on E and the set of closed positive partial forms with dense domain on E. The inverse bijection associates with a positive partial form $q$ the partial operator representing $q$.

By prop. 13, b) and 14, b), the mappings described in the statement are well-defined. Let us prove that they are inverse bijections of one another.

Let $u$ be a positive self-adjoint partial operator on E and $q$ the positive partial form associated with $u$. Let $v$ denote the positive self-adjoint operator representing $q$. Let $y\in$ dom($u$)$\subset$ dom($u^{1/2}$) $=$ dom($q$). For every $x\in$ dom($q$) $=$ dom($u^{1/2}$), we have

hence $y$ belongs to the domain of $v$ and satisfies $v(y) =u(y)$. The partial operator $v$ is therefore an extension of $u$; since $u$ and $v$ are self-adjoint, they are equal.

Conversely, let $q$ be a closed positive partial form with dense domain and $u$ the positive self-adjoint partial operator representing $q$. We have dom($u$)$\subset$ dom($u^{1/2}$). For $x$ and $y$ in dom($u$), we have

by prop. 13, a). Thus, the Hilbert spaces $E_q$ and $E_{q_u}$ both contain dom($u$) as a dense subspace (prop. 14, a) and prop. 13, b), respectively) and $\|x\|_q=\|x\|_{q_u}$ for every $x\in$ dom($u$). It follows that $E_q= E_{q_u}$ and that $q=q_u$ (lemma 8).