In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed

densely defined operator In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". ...

to have closed

range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

History

The theorem was proved by

Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...

in his

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

'' 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 In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...

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 = \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