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.


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


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:


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.

See also


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