Exercise 11, § 5

a) Let $q$ be a partial positive form on a complex Hilbert space E. Then $q$ is closed if and only if for every sequence $(x_n)$ of elements of dom($q$) which converges to $x$ in E and satisfies

lim$_{n,m\rightarrow+\infty}q(x_n-x_m, x_n-x_m) = 0$,

one has $x\in$ dom($q$) and $q(x_n-x, x_n-x)\rightarrow 0$.

b) Let $E = L^2(\mathbf{R}, \mu)$, where $\mu$ is Lebesgue measure. The positive Hermitian form $q$ with domain $\mathscr{D}(\mathbf{R})$ defined by $q(\varphi_1, \varphi_2) =\overline{\varphi_1(0)}\varphi_2(0)$ is not closed, and there exists no closed partial positive form extending $q$.