Difference between revisions of "Closed-graph theorem"
From Encyclopedia of Mathematics
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| + | Let $X$ 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Rudin, "Functional analysis" , McGraw-Hill (1979)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> W. Rudin, "Functional analysis" , McGraw-Hill (1979)</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge University Press (1964)</TD></TR> | ||
| + | </table> | ||
====Comments==== | ====Comments==== | ||
| − | Cf. also [[ | + | Cf. also [[Open-mapping theorem]] (also for the Banach homeomorphism theorem). |
Revision as of 18:13, 8 December 2014
Let $X$ 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).
How to Cite This Entry:
Closed-graph theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed-graph_theorem&oldid=35500
Closed-graph theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed-graph_theorem&oldid=35500
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article