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

Corollaries

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.