Exercise 3, § 4

Let $K =\mathbf{R}$ or $\mathbf{C}$. Let E, F be topological vector spaces over K and let $u$ be an operator with dense domain from E into F. Let $D'$ be the set of continuous linear forms $\lambda \in F'$ such that there exists a continuous linear form $\mu\in E'$ agreeing with $\lambda \circ u$ on the domain of $u$.

a) The set $D'$ is a vector subspace of $F'$, the linear form $\mu$ is uniquely determined by $\lambda$, and the mapping $\lambda \mapsto \mu$ of $D'$ into $E'$ is linear.

The partial operator from $F'$ into $E'$ whose domain is $D'$ and which is given by $\lambda \mapsto \mu$ for $\lambda \in D'$ is called the transpose of $u$, and is denoted by $^tu$.

b) Suppose that E and F are locally convex. The operator $u$ is closable if and only if the duality $F\times$ dom($^tu$)$\rightarrow K$ is separating.

c) Suppose that E and F are locally convex and that F is semireflexive. The operator $u$ is closable if and only if $^tu$ has dense domain.