Fredholm's theorem

Fredholm's theorem describes solvability for linear equations involving compact perturbations of the identity: the null space and the obstruction to solving the equation have equal finite dimension.

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Fredholm alternative: A solvability alternative for a linear equation and its adjoint: either the homogeneous equation has nonzero solutions, or compatibility conditions govern solvability. It states the kernel-versus-solvability dichotomy underlying Fredholm's theorem.
Linear operator: A map between vector spaces that preserves addition and scalar multiplication. Fredholm's theorem concerns equations defined by linear maps.
Fredholm integral equation: An integral equation whose unknown function appears both outside and inside an integral operator. Its solvability is a classical setting for Fredholm's theorem.
Erik Ivar Fredholm: A Swedish mathematician whose work on integral equations helped establish modern operator theory. His 1900 study gave the theorem its name and a foundational integral-equation form.
Lax–Milgram theorem: A theorem guaranteeing a unique weak solution for bounded coercive bilinear forms on Hilbert spaces. It ensures existence through coercivity rather than balancing kernel and cokernel dimensions.
Fredholm operator: A bounded linear operator between Banach spaces with finite-dimensional kernel, closed range, and finite-dimensional cokernel. The theorem's index conclusion applies to this class of operators.
Kernel (linear algebra): The set of inputs a linear map sends to zero. Its nonzero elements are precisely the solutions of the homogeneous equation.
Boundary value problem: A differential equation constrained by conditions imposed at the boundary of its domain. Many such problems reduce to operator equations with Fredholm solvability conditions.
Integral equation: An equation in which an unknown function appears under an integral sign. Fredholm first developed the solvability theory for this broader class of equations.
Banach fixed-point theorem: A theorem guaranteeing a unique fixed point for a contraction on a complete metric space. It proves uniqueness by contraction, not by the Fredholm alternative's range conditions.