The Closed Graph Theorem

For , Metric space, and , the graphh of is .

Note that continuous closed.

Theorem

, Banach Space, linear. Then continuous closed.

Proof

Trivial

Have closed in (with ) so is complete. Define by Want bounded; if Then so Now is bijective and is continuous (as a bounded linear map) We are done by Inversion Theorem.

Usecase

Normally, to show continuous we take and show . Now we have and if then .