View source for Closed-graph theorem

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.

====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>



====Comments====
Cf. also [[Open-mapping theorem|Open-mapping theorem]] (also for the Banach homeomorphism theorem).