Fredholm operator

A bounded linear operator between Banach spaces is Fredholm when its kernel and cokernel are finite-dimensional and its range is closed. Its index is the kernel dimension minus the cokernel dimension.

Kernel of a linear map: The subspace of inputs that a linear map sends to zero. Its finite dimension is the first term in the Fredholm index.
Atkinson's theorem: A bounded operator on a Banach space is Fredholm exactly when it is invertible modulo compact operators. It characterizes Fredholmness through an inverse up to compact errors.
Elliptic partial differential equation: A partial differential equation whose highest-order terms have an elliptic symbol. Elliptic boundary problems often define Fredholm maps between function spaces.
Invertible operator: A linear operator with a two-sided inverse. Invertibility forces both kernel and cokernel to vanish, a stronger condition.
Image of a linear map: The set of outputs attained by a map on its domain. Fredholm operators require this subspace to be closed.
Compact operator: A linear operator that maps bounded sets to sets with compact closure. Compact perturbations preserve Fredholmness and leave the index unchanged.
Boundary value problem: A differential equation supplemented by conditions on the boundary of its domain. Fredholmness captures finite-dimensional obstructions to solving such problems.
Semi-Fredholm operator: An operator with closed range and at least one of finite-dimensional kernel or finite-codimensional range. It relaxes Fredholmness by requiring only one of the two finiteness conditions.
Cokernel: The quotient of a codomain by the image of a linear map. Its finite dimension measures how many independent output directions are missing.
Fredholm alternative: A solvability principle linking a linear equation to the kernel of its adjoint. It turns Fredholm structure into solvability conditions for operator equations.

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