In the mathematical theory of

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

s, the closed range theorem gives necessary and sufficient conditions for a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

densely defined operator to have

range.

History

The theorem was proved by Stefan Banach in his

1932 Events January * January 4 – The British authorities in India arrest and intern Mahatma Gandhi and Vallabhbhai Patel. * January 9 – Sakuradamon Incident (1932), Sakuradamon Incident: Korean nationalist Lee Bong-chang fails in his effort ...

'' 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

X.

R(T)  = N(T')^\perp = \left\.

R(T') = N(T)^\perp  = \left\.

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.

References

* * . {{Functional Analysis Banach spaces Theorems in functional analysis