Lemma 2

Let E, F and G be normed spaces over K, F being a Banach space. Let $u$ be a closed partial operator from E into F and let $v\in \mathscr{L}(F; G)$. If there exists $C\in \mathbf{R}_+$ such that

for every $x\in$ dom($v\circ u$) $=$ dom($u$), that is to say, if the linear mapping $x\mapsto u(x)$ from $E_{v\circ u}$ into F is continuous, then $v\circ u$ is closed.

Let us write $w=v\circ u$. Let $(x_n, w(x_n))_{n\in\mathbf{N}}$ be a sequence in the graph of $w$ which converges in $E\times G$. Let $x$ be the limit of the sequence $(x_n)$. The hypothesis implies that the sequence $(u(x_n))_{n\in\mathbf{N}}$ is then a Cauchy sequence in F; it converges to an element $y$ of F. Since $u$ is closed, one has therefore $x\in$ dom($u$) and $y=u(x)$. Then $w(x_n) =v(u(x_n))$ tends to $v(y) =$ $v(u(x))$ since $v$ is continuous, hence the graph of $w$ is closed.

Let $u$ be a closed partial operator on a Banach space E and F a subspace of dom($u$). If F is dense in the Banach space $E_u$, then the restriction of $u$ to F is closable, and its closure is equal to $u$. One then says that F is a core for $u$.