Example

Let $u$ be a positive partial operator on E which is not necessarily closed. One defines a positive partial form $q$ with domain dom($u$) by

for $x$ and $y$ in dom($u$). Let $E_q$ be the separated pre-Hilbert space of Definition 9 of IV, p. 292. Let $\widetilde{E}_q$ be the Hilbert space completion of $E_q$, whose scalar product is again denoted by $(x, y)\mapsto (x|y)_q$. Since the canonical injection $\iota$ of $E_q$ into E is continuous, it admits a unique continuous extension, denoted by $\widetilde{\iota}$, from $\widetilde{E}_q$ into E. The Hermitian form $q$ is continuous on $E_q$, and therefore is also extended to a unique continuous positive Hermitian form $\widetilde{q}$ on $\widetilde{E}_q$.

Let us prove that the linear mapping $\widetilde{\iota}$ is injective. Let $x\in$ Ker($\widetilde{\iota}$). Consider a sequence $(x_n)_{n\in\mathbf{N}}$ in $E_q$ which converges to $x$ in $\widetilde{E}_q$. Then one has

lim$_{n\rightarrow+\infty}\iota (x_n) =\widetilde{\iota}(x) = 0$,

hence the sequence $(x_n)_{n\in\mathbf{N}}$ converges to 0 in E. Let $y\in E_q$. Since $\widetilde{q}$ is continuous on $\widetilde{E}_q$, it follows that

$(x|y)_q=$ lim$_{n\rightarrow+\infty}(x_n|y)_q=$ lim$_{n\rightarrow+\infty}(\langle x_n|y\rangle +q(x_n, y))$

= lim$_{n\rightarrow+\infty}(\langle x_n|y\rangle +\langle x_n|u(y)\rangle )= 0$.

Since the space $E_q$ is dense in $\widetilde{E}_q$, it follows that $x= 0$, as desired.

Identifying $\widetilde{E}_q$ with its image under $\widetilde{\iota}$ in E, one interprets $\widetilde{q}$ as a closed positive partial form with dense domain extending $q$. The positive self-adjoint operator associated with $\widetilde{q}$ is a self-adjoint extension of $u$. It is called the Friedrichs extension of $u$.