Closed-graph theorem
Let and Y be complete metric linear spaces with translation-invariant metrics, i.e. \rho_X(x_1+a,x_2+a) = \rho_X(x_1,x_2), x_1,x_2,a \in X (similarly for Y), and let T be a linear operator from X to Y. If the graph \mathrm{Gr}(T) = \{ (x,Tx) : x \in X \} of this operator is a closed subset of the Cartesian product X \times Y, then T 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 University Press (1964) |
Comments
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=35501