In
mathematics
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 ...
, Fredholm's theorems are a set of celebrated results of
Ivar Fredholm in the
Fredholm theory of
integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of
linear algebra, or in terms of the
Fredholm operator on
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
Fredholm alternative is one of the Fredholm theorems.
Linear algebra
Fredholm's theorem in linear algebra is as follows: if ''M'' is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, then the
orthogonal complement of the
row space of ''M'' is the
null space of ''M'':
:
Similarly, the orthogonal complement of the column space of ''M'' is the null space of the adjoint:
:
Integral equations
Fredholm's theorem for integral equations is expressed as follows. Let
be an
integral kernel
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in t ...
, and consider the
homogeneous equations
:
and its complex adjoint
:
Here,
denotes the
complex conjugate of the
complex number , and similarly for
. Then, Fredholm's theorem is that, for any fixed value of
, these equations have either the trivial solution
or have the same number of
linearly independent solutions
,
.
A sufficient condition for this theorem to hold is for
to be
square integrable on the rectangle