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.

History

The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Statement

Let $X$ and $Y$ be Banach spaces, $T\colon D(T)\to Y$ 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 $X$ ,
$R(T)=N(T')^{\perp }=\{y\in Y|\langle x^{*},y\rangle =0\quad {{\text{for all}}}\quad x^{*}\in N(T')\}$ ,
$R(T')=N(T)^{\perp }=\{x^{*}\in X'|\langle x^{*},y\rangle =0\quad {{\text{for all}}}\quad y\in N(T)\}$ .

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.

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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Closed range theorem

History

Statement

Corollaries

See also

References