Fredholm theory

Fredholm theory studies linear operators with finite-dimensional kernels and cokernels and closed range. Its central invariant, the Fredholm index, measures the difference between kernel and cokernel dimensions and governs solvability and stability.

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Bounded linear operator: A linear map between normed vector spaces whose output changes by at most a constant multiple of the input norm. Fredholm theory typically studies bounded maps between Banach spaces.
Fredholm alternative: A solvability theorem relating a linear operator equation to the homogeneous equation for its adjoint. It turns Fredholm structure into precise conditions for solving operator equations.
Elliptic operator: A differential operator whose principal symbol is invertible away from the zero section. Elliptic operators on compact spaces commonly define Fredholm maps between suitable function spaces.
Invertible operator: A linear operator that is bijective and has a bounded inverse in the relevant normed spaces. Fredholm operators may fail to be invertible while retaining only finite-dimensional defects.
Kernel (linear algebra): The set of inputs a linear map sends to zero. Its finite dimension is one part of the Fredholm index.
Riesz–Schauder theory: The theory describing spectral properties of compact operators on infinite-dimensional spaces. It supplies foundational results for operators that are identity plus compact.
Elliptic boundary value problem: An elliptic differential equation supplemented by boundary conditions designed to produce well-posed behavior. Its solvability and finite-dimensional obstructions are analyzed using Fredholm operators.
Semi-Fredholm operator: An operator with closed range and either finite-dimensional kernel or finite-dimensional cokernel. It relaxes one of the two finiteness conditions required for Fredholmness.
Cokernel: The quotient of a linear map's codomain by its image. Its finite dimension supplies the other part of the index.
Compact operator: A linear operator that maps bounded sets to sets with compact closure. Compact perturbations preserve Fredholmness and leave the index unchanged.