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 Erik Ivar Fredholm (7 April 1866 – 17 August 1927) was a Swedish mathematician whose work on integral equations and operator theory foreshadowed the theory of Hilbert spaces. Biography Fredholm was born in Stockholm in 1866. He obtained his P ...

in the

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 ...

integral equations In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...

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

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

, or in terms of the

Fredholm operator In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : '' ...

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 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 ...

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'' (franchis ...

, then the

orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

of the

row space Row or ROW may refer to: Exercise *Rowing, or a form of aquatic movement using oars *Row (weight-lifting), a form of weight-lifting exercise Math *Row vector, a 1 × ''n'' matrix in linear algebra. *Row (database), a single, implicitly structured ...

of ''M'' is the

null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...

of ''M'': :

(\operatorname M)^\bot = \ker M.

Similarly, the orthogonal complement of the column space of ''M'' is the null space of the adjoint: :

(\operatorname M)^\bot = \ker M^*.

Integral equations

Fredholm's theorem for integral equations is expressed as follows. Let

K(x,y)

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 equation In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three ...

s :

\int_a^b K(x,y) \phi(y) \,dy = \lambda \phi(x)

and its complex adjoint :

\int_a^b \psi(x) \overline \, dx = \overline \psi(y).

Here,

\overline

denotes the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

of the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

\lambda

, and similarly for

\overline

. Then, Fredholm's theorem is that, for any fixed value of

\lambda

, these equations have either the trivial solution

\psi(x)=\phi(x)=0

or have the same number of

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

solutions

\phi_1(x),\cdots,\phi_n(x)

\psi_1(y),\cdots,\psi_n(y)

. A sufficient condition for this theorem to hold is for

K(x,y)

to be

square integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...

on the rectangle

,b times,b /math> (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

, this result generalizes to

integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms invol ...

s on multi-dimensional spaces, including, for example,

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

Existence of solutions

One of Fredholm's theorems, closely related to the

, concerns the existence of solutions to the inhomogeneous

Fredholm equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to sol ...

\lambda \phi(x)-\int_a^b K(x,y) \phi(y) \,dy=f(x).

Solutions to this equation exist if and only if the function

f(x)

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

to the complete set of solutions

\

of the corresponding homogeneous adjoint equation: :

\int_a^b \overline f(x) \,dx=0

where

\overline

is the complex conjugate of

\psi_n(x)

and the former is one of the complete set of solutions to :

\lambda\overline -\int_a^b \overline K(x,y) \,dx=0.

A sufficient condition for this theorem to hold is for

K(x,y)

to be

on the rectangle

,b times,b /math>.

References

* E.I. Fredholm, "Sur une classe d'equations fonctionnelles", ''Acta Math.'', 27 (1903) pp. 365–390. * * {{Functional analysis Fredholm theory Linear algebra Theorems in functional analysis