# 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.

Let $X$ and $Y$ be Banach spaces, $T\colon D(T) \to Y$ 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 $X$,
• $R(T) = N(T')^\perp=\{y\in Y | \langle x^*,y\rangle = 0\quad {\text{for all}}\quad x^*\in N(T')\}$,
• $R(T') = N(T)^\perp=\{x^*\in X' | \langle x^*,y\rangle = 0\quad {\text{for all}}\quad y\in N(T)\}$.

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.