Definition 43.8.1 (Closed).label Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a linear operator, then $T$ is closed if its graph
\[\Gamma(T) = \bracsn{(x, y) \in D(T) \times H| Tx = y}\]
is a closed subspace of $H \times H$.
The operator $T$ is closable if it admits a closed extension $S \supset T$.
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