Closed range theorem

Short description: Mathematical theorem about Banach spaces

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, $T:D(T)\to Y$ a closed linear operator whose domain $D(T)$ is dense in $X,$ and $T^{\prime }$ the transpose of $T$. The theorem asserts that the following conditions are equivalent:

• $R(T),$ the range of $T,$ is closed in $Y.$
• $R(T^{\prime }),$ the range of $T^{\prime },$ is closed in $X^{\prime },$ the dual of $X.$
• $R(T)=N(T^{\prime }{)}^{\perp }=\{y\in Y:⟨x^{*},y⟩=0\text{for all}{x}^{*}\in N(T^{\prime })\}.$
• $R(T^{\prime })=N(T{)}^{\perp }=\{x^{*}\in X^{\prime }:⟨x^{*},y⟩=0\text{for all}y\in N(T)\}.$

Where $N(T)$ and $N(T^{\prime })$ are the null space of $T$ and $T^{\prime }$, respectively.

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 $T^{\prime }$ has a continuous inverse. Similarly, $R(T^{\prime })=X^{\prime }$ if and only if $T$ has a continuous inverse.

References

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