Definition:Closed Densely-Defined Linear Operator

Definition
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $\struct {\map D T, T}$ be a densely defined linear operator on $\HH$.

Let $\struct {\HH \times \HH, \norm \cdot_{\HH \times \HH} }$ be the direct product $\HH \times \HH$ equipped with the direct product norm.

We say that $\struct {\map D T, T}$ is closed if:


 * $\set {\tuple {x, T x} \in \HH \times \HH : x \in \map D T}$

is a closed subset of $\HH \times \HH$.