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 .