Closed range theorem

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(X) → Y a closed linear operator whose domain D(X) is dense in X, and $\scriptstyle {T'}$ its transpose. The theorem asserts that the following conditions are equivalent:

• R(T), the range of T, is closed in Y
• $\scriptstyle {R(T')}$, the range of $\scriptstyle {T'}$, is closed in $\scriptstyle {X'}$, the dual of X
• $R(T)=N(T')^{\perp }=\{y\in Y|\langle x^{*},y\rangle =0\quad {\text{for all}}\quad x^{*}\in N(T')\}$
• $R(T')=N(T)^{\perp }=\{x^{*}\in X'|\langle x^{*},y\rangle =0\quad {\text{for all}}\quad y\in N(T)\}$

When T is injective, the above conditions are equivalent to that the inverse of $T:D(X)\to {\overline {R(T)}}$ (which is only densely defined in general) is defined everywhere (thus, bounded by the closed graph theorem).

References

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