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. Linear algebra 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 s.t. AT y = 0, yTb = 0 . Integral equations 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. Functional analysis Results on the Fredholm operator generalize these results to vector spaces of infinite dimensions, 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, T induces a linear operator acting on a Banach space V of functions , which we also call T, so that is 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. The operator K given by convolution with an L2