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Concept
Fredholm alternative
Summary
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. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
If V is an n-dimensional vector space and is a linear transformation, then exactly one of the following holds:
For each vector v in V there is a vector u in V so that . In other words: T is surjective (and so also bijective, since V is finite-dimensional).
A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:
Either: A x = b has a solution x
Or: AT y = 0 has a solution y with yTb ≠ 0.
In other words, A x = b has a solution if and only if for any y such that AT y = 0, it follows that yTb = 0 .
Let be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,
and the inhomogeneous equation
The Fredholm alternative is the statement that, for every non-zero fixed complex number , either the first equation has a non-trivial solution, or the second equation has a solution for all .
A sufficient condition for this statement to be true is for to be square integrable on the rectangle (where a and/or b may be minus or plus infinity). The integral operator defined by such a K is called a Hilbert–Schmidt integral operator.
Results about Fredholm operators generalize these results to complete normed vector spaces of infinite dimensions; that is, Banach spaces.
The integral equation can be reformulated in terms of operator notation as follows. Write (somewhat informally)
to mean
with the Dirac delta function, considered as a distribution, or generalized function, in two variables.
Then by convolution, induces a linear operator acting on a Banach space of functions
given by
with given by
In this language, the Fredholm alternative for integral equations is seen to be analogous to the Fredholm alternative for finite-dimensional linear algebra.
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