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

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

## 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.