Let and be complete metric linear spaces with translation-invariant metrics, i.e. , (similarly for ), and let be a linear operator from to . If the graph of this operator is a closed subset of the Cartesian product , then is continuous. The closed-graph theorem has various generalizations; for example: a linear mapping with closed graph from a separable barrelled space into a perfectly-complete space is continuous. Closely related theorems are the open-mapping theorem and Banach's homeomorphism theorem.

References

Cf. also Open-mapping theorem (also for the Banach homeomorphism theorem).