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.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.
