# 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 ,
- ,
- .

## 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.

## See also

## References

- Yosida, K. (1980),
*Functional Analysis*, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag.

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