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, T : D(T) \to 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')^\perp = \left\{ y \in Y : \langle x^*,y \rangle = 0 \quad {\text{for all}}\quad x^* \in N(T') \right\}.
  • R(T') = N(T)^\perp = \left\{x^* \in X' : \langle x^*,y \rangle = 0 \quad {\text{for all}}\quad y \in N(T) \right\}.

Where N(T) and N(T') are the null space of T and T', respectively.

Note that there is always an inclusion R(T)\subseteq N(T')^\perp, because if y=Tx and x^*\in N(T'), then \langle x^*,y\rangle = \langle T'x^*,x\rangle = 0. Likewise, there is an inclusion R(T')\subseteq N(T)^\perp. 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 X \overset{p}\to X/\operatorname{ker}T \overset{T_0}\to \operatorname{im}T \overset{i}\hookrightarrow Y. Dually, T' is

Y' \to (\operatorname{im}T)' \overset{T_0'}\to (X/\operatorname{ker}T)' \to X'.

Now, if \operatorname{im}T 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 \operatorname{im}(T') = \operatorname{ker}(T)^{\bot}. (More work is needed for the other implications.) \square

Further reading

References

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