Closed range theorem

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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:

  • , the range of , is closed in ,
  • , the range of , is closed in , the dual of ,
  • ,
  • .


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[edit]


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