Closed Graph Theorem

The closed graph theorem states that a linear operator between two Banach spaces  and  is continuous iff it has a closed graph, where the "graph"  is considered closed if it is a closed subset of  equipped with the product topology.

The closed graph theorem also holds for Fréchet spaces.

REFERENCES:

Conway, J. B. A Course in Functional Analysis. New York: Springer-Verlag, 1990.

Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.