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.
|||W. Rudin, "Functional analysis" , McGraw-Hill (1979)|
|||A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964)|
Cf. also Open-mapping theorem (also for the Banach homeomorphism theorem).
Closed-graph theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed-graph_theorem&oldid=11648