Fredholm Alternative

The Fredholm Alternative, named for the Swedish mathematician Ivar Fredholm, is a theorem of functional analysis that can be stated equivalently in linear algebra, in the theory of integral equations, or for compact operators on a Hilbert or Banach space. It shows that for such an operator, either an associated linear equation has a unique solution for every choice of right-hand side, or the corresponding homogeneous equation has a nontrivial solution, so that solvability comes down to a clear either-or choice. Part of the result establishes that every nonzero complex number in the spectrum of a compact operator must be an eigenvalue, and the theorem underlies much of the classical theory of integral equations.