View source for Open-mapping theorem

A continuous [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683001.png" /> mapping a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683002.png" /> onto all of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683003.png" /> is an [[Open mapping|open mapping]], i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683004.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683005.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683006.png" /> which is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683007.png" />. This was proved by S. Banach. Furthermore, a continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683008.png" /> giving a one-to-one transformation of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o0683009.png" /> onto a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830010.png" /> is a [[Homeomorphism|homeomorphism]], i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830011.png" /> is also a continuous linear operator (Banach's homeomorphism theorem).

The conditions of the open-mapping theorem are satisfied, for example, by every non-zero continuous linear functional defined on a real (complex) Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830012.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830013.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830014.png" />).

The open-mapping theorem can be generalized as follows: A continuous linear operator mapping a fully-complete (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830015.png" />-complete) [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830016.png" /> onto a [[Barrelled space|barrelled space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068300/o06830017.png" /> is an open mapping. The [[Closed-graph theorem|closed-graph theorem]] can be considered alongside with the open-mapping theorem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)</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====
A recent comprehensive study of the closed-graph theorem can be found in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. de Wilde,   "Closed graph theorems and webbed spaces" , Pitman  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Jarchow,   "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR></table>