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
and
be Banach spaces,
a closed linear operator whose domain
is dense in
and
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 ...
of
. The theorem asserts that the following conditions are equivalent:
*
the range of
is closed in
*
the range of
is closed in
the
dual of
*
*
Where
and
are the null space of
and
, respectively.
Corollaries
Several corollaries are immediate from the theorem. For instance, a densely defined closed operator
as above has
if and only if the transpose
has a continuous inverse. Similarly,
if and only if
has a continuous inverse.
References
*
* .
{{Functional Analysis
Banach spaces
Theorems in functional analysis