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.

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.) $\square$

References

Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 11 January 2014. Retrieved 11 July 2020.
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

Statement

Corollaries

Sketch of proof

Further reading

References