1. History

2. Statement

3. Corollaries

4. See also

5. References

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

• Fundamental theorem of linear algebra

References

• {{Citation | last1=Yosida | first1=K. | title=Functional Analysis | publisher=Springer-Verlag | location=Berlin, New York | edition=6th | series=Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 | year=1980}}.

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