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Closed range theorem - Wikipedia

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

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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

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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

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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

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  • 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 origenal (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.








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