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 SpacePortions of this entry contributed by Mohammad Sal Moslehian
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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.Referenced on Wolfram|Alpha
Closed Graph TheoremCite this as:
Moslehian, Mohammad Sal and Weisstein, Eric W. "Closed Graph Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedGraphTheorem.html