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 X and Y be Banach spaces, T : D ( T ) → Y a closed linear operator whose domain D ( T ) is dense in X , and T ′ the transpose of T . The theorem asserts that the following conditions are equivalent:
R ( T ) , the range of T , is closed in Y , R ( T ′ ) , the range of T ′ , is closed in X ′ , the dual of X , R ( T ) = N ( T ′ ) ⊥ = { y ∈ Y | ⟨ x ∗ , y ⟩ = 0 for all x ∗ ∈ N ( T ′ ) } , R ( T ′ ) = N ( T ) ⊥ = { x ∗ ∈ X ′ | ⟨ x ∗ , y ⟩ = 0 for all y ∈ N ( T ) } .Several corollaries are immediate from the theorem. For instance, a densely defined closed operator T as above has R ( T ) = Y if and only if the transpose T ′ has a continuous inverse. Similarly, R ( T ′ ) = X ′ if and only if T has a continuous inverse.
