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.