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 edit
The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.
Statement edit
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 edit
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 edit
- 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.