# Closed-graph theorem

From Encyclopedia of Mathematics

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

[1] | W. Rudin, "Functional analysis" , McGraw-Hill (1979) |

[2] | A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964) |

#### Comments

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

**How to Cite This Entry:**

Closed-graph theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Closed-graph_theorem&oldid=11648

This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article