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.

Portions of this entry contributed by Mohammad Sal Moslehian

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Moslehian, Mohammad Sal and Weisstein, Eric W. "Closed Graph Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClosedGraphTheorem.html