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

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 has a continuous inverse. Similarly, $\scriptstyle{R(T') = X'}$ if and only if T has a continuous inverse.

See also

• Fundamental theorem of linear algebra

References

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

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