Closed range theorem
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.
History
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.
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.
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