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
:
D
(
T
)
→
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
′
)
⊥
=
{
y
∈
Y
|
⟨
x
∗
,
y
⟩
=
0
for all
x
∗
∈
N
(
T
′
)
}
,
R
(
T
′
)
=
N
(
T
)
⊥
=
{
x
∗
∈
X
′
|
⟨
x
∗
,
y
⟩
=
0
for all
y
∈
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.
