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.
be Banach spaces,
a closed linear operator whose domain
is dense in
the transpose of
The theorem asserts that the following conditions are equivalent: Where
are the null space of
Note that there is always an inclusion
Likewise, there is an inclusion
So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.
Several corollaries are immediate from the theorem.
For instance, a densely defined closed operator
has a continuous inverse.
has a continuous inverse.
Since the graph of T is closed, the proof reduces to the case when
is a bounded operator between Banach spaces.
factors as
im
{\displaystyle X{\overset {p}{\to }}X/\operatorname {ker} T{\overset {T_{0}}{\to }}\operatorname {im} T{\overset {i}{\hookrightarrow }}Y}
im
{\displaystyle \operatorname {im} T}
is closed, then it is Banach and so by the open mapping theorem,
is a topological isomorphism.
is an isomorphism and then
im (
{\displaystyle \operatorname {im} (T')=\operatorname {ker} (T)^{\bot }}
(More work is needed for the other implications.)