Closed range theorem: Difference between revisions
TakuyaMurata (talk | contribs) add a sketch of proof Tag: Disambiguation links added |
TakuyaMurata (talk | contribs) →Sketch of proof: just note the proof is not complete |
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Since the graph of ''T'' is closed, the proof reduces to the case when <math>T : X \to Y</math> is a bounded operator between Banach spaces. Now, <math>T</math> factors as <math>X \overset{p}\to X/\operatorname{ker}T \overset{T_0}\to \operatorname{im}T \overset{i}\hookrightarrow Y</math>. Dually, <math>T'</math> is |
Since the graph of ''T'' is closed, the proof reduces to the case when <math>T : X \to Y</math> is a bounded operator between Banach spaces. Now, <math>T</math> factors as <math>X \overset{p}\to X/\operatorname{ker}T \overset{T_0}\to \operatorname{im}T \overset{i}\hookrightarrow Y</math>. Dually, <math>T'</math> is |
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:<math>Y' \to (\operatorname{im}T)' \overset{T_0'}\to (X/\operatorname{ker}T)' \to X'.</math> |
:<math>Y' \to (\operatorname{im}T)' \overset{T_0'}\to (X/\operatorname{ker}T)' \to X'.</math> |
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Now, if <math> \operatorname{im}T</math> is closed, then it is Banach and so by the [[open mapping theorem]], <math>T_0</math> is a topological isomorphism. It follows that <math>T_0'</math> is an isomorphism and then <math>\operatorname{im}(T') = \operatorname{ker}(T)^{\bot}</math>. <math>\square</math> |
Now, if <math> \operatorname{im}T</math> is closed, then it is Banach and so by the [[open mapping theorem]], <math>T_0</math> is a topological isomorphism. It follows that <math>T_0'</math> is an isomorphism and then <math>\operatorname{im}(T') = \operatorname{ker}(T)^{\bot}</math>. (More work is needed for the other implications.<!-- and we should add proofs for those too. -->) <math>\square</math> |
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==References== |
==References== |
Revision as of 13:08, 17 July 2024
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.
The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.
Statement
Let and be Banach spaces, a closed linear operator whose domain is dense in and the transpose of . The theorem asserts that the following conditions are equivalent:
- the range of is closed in
- the range of is closed in the dual of
Where and are the null space of and , respectively.
Note that there is always an inclusion , because if and , then . Likewise, there is an inclusion . So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.
Corollaries
Several corollaries are immediate from the theorem. For instance, a densely defined closed operator as above has if and only if the transpose has a continuous inverse. Similarly, if and only if has a continuous inverse.
Sketch of proof
Since the graph of T is closed, the proof reduces to the case when is a bounded operator between Banach spaces. Now, factors as . Dually, is
Now, if is closed, then it is Banach and so by the open mapping theorem, is a topological isomorphism. It follows that is an isomorphism and then . (More work is needed for the other implications.)
References
- Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
- Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag.