Closed range theorem

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

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Banach spaces, ${\displaystyle T\colon D(T)\to Y}$ a closed linear operator whose domain ${\displaystyle D(T)}$ is dense in ${\displaystyle X}$, and ${\displaystyle T'}$ the transpose of ${\displaystyle T}$. The theorem asserts that the following conditions are equivalent:

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

Where ${\displaystyle N(T)}$ and ${\displaystyle N(T')}$ are the null space of ${\displaystyle T}$ and ${\displaystyle T'}$, respectively.

Several corollaries are immediate from the theorem. For instance, a densely defined closed operator ${\displaystyle T}$ as above has ${\displaystyle R(T)=Y}$ if and only if the transpose ${\displaystyle T'}$ has a continuous inverse. Similarly, ${\displaystyle R(T')=X'}$ if and only if ${\displaystyle T}$ has a continuous inverse.

• Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
• Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag.