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Difference between revisions of "Open-mapping theorem"

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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).
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A continuous [[Linear operator|linear operator]] $A$ mapping a [[Banach space|Banach space]] $X$ onto all of a Banach space $Y$ is an [[Open mapping|open mapping]], i.e. $A(G)$ is open in $Y$ for any $G$ which is open in $X$. This was proved by S. Banach. Furthermore, a continuous linear operator $A$ giving a one-to-one transformation of a Banach space $X$ onto a Banach space $Y$ is a [[Homeomorphism|homeomorphism]], i.e. $A^{-1}$ 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" />).
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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 $X$ with values in $\mathbf R$ (in $\mathbf C$).
  
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.
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The open-mapping theorem can be generalized as follows: A continuous linear operator mapping a fully-complete (or $B$-complete) [[Topological vector space|topological vector space]] $X$ onto a [[Barrelled space|barrelled space]] $Y$ is an open mapping. The [[Closed-graph theorem|closed-graph theorem]] can be considered alongside with the open-mapping theorem.
  
 
====References====
 
====References====

Latest revision as of 15:25, 27 August 2014

A continuous linear operator $A$ mapping a Banach space $X$ onto all of a Banach space $Y$ is an open mapping, i.e. $A(G)$ is open in $Y$ for any $G$ which is open in $X$. This was proved by S. Banach. Furthermore, a continuous linear operator $A$ giving a one-to-one transformation of a Banach space $X$ onto a Banach space $Y$ is a homeomorphism, i.e. $A^{-1}$ 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 $X$ with values in $\mathbf R$ (in $\mathbf C$).

The open-mapping theorem can be generalized as follows: A continuous linear operator mapping a fully-complete (or $B$-complete) topological vector space $X$ onto a barrelled space $Y$ is an open mapping. The closed-graph theorem can be considered alongside with the open-mapping theorem.

References

[1] K. Yosida, "Functional analysis" , Springer (1980)
[2] A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964)


Comments

A recent comprehensive study of the closed-graph theorem can be found in [a1].

References

[a1] M. de Wilde, "Closed graph theorems and webbed spaces" , Pitman (1978)
[a2] H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[a3] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
How to Cite This Entry:
Open-mapping theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Open-mapping_theorem&oldid=13271
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article