Closed range theorem
# 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, $T:D(T)\to Y$ 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.

## 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.

