In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory

Fredholm theory

In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm...

 of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, or in terms of the Fredholm operator

Fredholm operator

In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....

 on Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s.

The Fredholm alternative

Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators...

 is one of the Fredholm theorems.

Linear algebra

Fredholm's theorem in linear algebra is as follows: if M is a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, then the orthogonal complement of the row space

Row space

In linear algebra, the row space of a matrix is the set of all possible linear combinations of its row vectors. The row space of an m × n matrix is a subspace of n-dimensional Euclidean space...

 of M is the null space

Null space

In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean 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, and consider the homogeneous equation

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

s


and its complex adjoint


Here, denotes the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

 of the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 , 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 (where a and/or b may be minus or plus infinity).

Here, the integral is expressed as a one-dimensional integral on the real number line. In Fredholm theory

Fredholm theory

In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm...

, this result generalizes to integral operators on multi-dimensional spaces, including, for example, Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s.

Existence of solutions

One of the Fredholm theorem's closely related to the Fredholm alternative

Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators...

, concerns the existence of solutions to the inhomogeneous Fredholm equation


Solutions to this equation exist if and only if the function is orthogonal to the complete set of solutions of the corresponding homogeneous adjoint equation:


where is the complex conjugate of and the former is one of the complete set of solutions to


A sufficient condition for this theorem to hold is for to be square integrable on the rectangle .