Closed Graph Theorem

The closed graph theorem states that a linear operator between two Banach spaces X and Y is continuous iff it has a closed graph, where the "graph" {(x,f(x)):x in X} is considered closed if it is a closed subset of X×Y equipped with the product topology.

The closed graph theorem also holds for Fréchet spaces.

See also

Banach Space

Portions of this entry contributed by Mohammad Sal Moslehian

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Closed Graph Theorem

Cite this as:

Moslehian, Mohammad Sal and Weisstein, Eric W. "Closed Graph Theorem." From MathWorld--A Wolfram Web Resource.

Subject classifications