Mathematical theorem about Banach spaces
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.