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.

Statement

the transpose. The theorem asserts that the following conditions are equivalent:

• the dual

, respectively.

. So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.

Corollaries

has a continuous inverse.

Sketch of proof

Since the graph of T is

is closed, then it is Banach and so by the open mapping theorem. (More work is needed for the other implications.

Further reading

• https://nalab.mind.meiji.ac.jp/~mk/labo/text/functional-analysis-3/node22.html#jtheorem:________________________ in Japanese

References

• .