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Difference between revisions of "Closed-graph theorem"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225402.png" /> be complete metric linear spaces with translation-invariant metrics, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225404.png" /> (similarly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225405.png" />), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225406.png" /> be a linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225407.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225408.png" />. If the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c0225409.png" /> of this operator is a closed subset of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c02254010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022540/c02254011.png" /> 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.
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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 Univ. Press  (1964)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1979)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Robertson,  W.S. Robertson,  "Topological vector spaces" , Cambridge University Press  (1964)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
Cf. also [[Open-mapping theorem|Open-mapping theorem]] (also for the Banach homeomorphism theorem).
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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
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article