Closed Graph Theorem

Closed Graph Theorem

An (everywhere-defined) linear operator between Banach spaces is bounded iff its graph is closed.

We prove a slightly more general version:

Closed Graph Theorem (Variant)

Let and be Banach spaces and a linear operator with domain closed in . Then is bounded if and only if its graph is closed.

Proof   Let us assume first that is bounded. Let be a sequence in that converges to some element . This means that and for . The continuity of implies . Since a convergent series in a Hausdorff space has a unique limit, it follows that ; hence lies in . This shows that is closed.

Conversely, suppose that is a closed subspace of . Note that is a Banach space with norm . Therefore, is itself as Banach space in the restricted norm . The canonical projections and are bounded. Clearly, is bijective, so its inverse is a bounded operator by the Bounded Inverse Theorem. Consequently, the composition, is bounded. To complete the proof, observe that .