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In the [[mathematics|mathematical]] theory of [[Banach space]]s, the '''closed range theorem''' gives necessary and sufficient conditions for a [[closed linear operator|closed]] [[densely defined operator]] to have [[closed set|closed]] [[range of a function|range]]. The theorem was proved by [[Stefan Banach]] in his 1932 ''Théorie des opérations linéaires''.
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Let <math>X</math> and <math>Y</math> be Banach spaces, <math>T\colon D(T) \to Y</math> a closed linear operator whose domain <math>D(T)</math> is dense in <math>X</math>, and <math>T'</math> the [[Unbounded_operator#Transpose|transpose]] of <math>T</math>. The theorem asserts that the following conditions are equivalent:
 
* <math>R(T)</math>, the range of <math>T</math>, is closed in <math>Y</math>,
* <math>R(T')</math>, the range of <math>T'</math>, is closed in <math>X'</math>, the [[continuous dual space|dual]] of <math>X</math>,
* <math>R(T) = N(T')^\perp=\{y\in Y | \langle x^*,y\rangle = 0\quad {\text{for all}}\quad x^*\in N(T')\}</math>,
* <math>R(T') = N(T)^\perp=\{x^*\in X' | \langle x^*,y\rangle = 0\quad {\text{for all}}\quad y\in N(T)\}</math>.
 
Several corollaries are immediate from the theorem. For instance, a densely defined closed operator <math>T</math> as above  has <math>R(T)=Y</math> if and only if the transpose <math>T'</math> has a continuous inverse. Similarly, <math>R(T') = X'</math> if and only if <math>T</math> has a continuous inverse.
 
== See also ==
* [[Fundamental theorem of linear algebra]]
 
==References==
*{{Citation | last1=Yosida | first1=K. | title=Functional Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=6th | series=Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 | year=1980}}.
 
{{Functional Analysis}}
 
[[Category:Theorems in functional analysis]]
 
{{mathanalysis-stub}}

Latest revision as of 12:28, 5 May 2014

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