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

Let X and Y be Banach spaces, 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: Where N(T) and N(T') are the null space of T and T', respectively. Note that there is always an inclusion, because if y=Tx and , then. Likewise, there is an inclusion. So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.

Corollaries

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.

Sketch of proof

Since the graph of T is closed, the proof reduces to the case when T : X \to Y is a bounded operator between Banach spaces. Now, T factors as. Dually, T' is Now, if is closed, then it is Banach and so by the open mapping theorem, T_0 is a topological isomorphism. It follows that T_0' is an isomorphism and then. (More work is needed for the other implications.) \square