A bounded linear map between Banach spaces whose kernel and cokernel are finite-dimensional and whose range is closed; its index is dim(kernel) minus dim(cokernel).

Principle

Fredholm property is stable under compact perturbations and the index is an integer invariant under such perturbations.

Demonstration

An integral operator on L^2 with a smooth kernel defines a Fredholm operator: its kernel and cokernel are finite-dimensional and the index can be computed from boundary data.

Misapplication

Calling any bounded operator with closed range a Fredholm operator without checking finite-dimensionality of kernel or cokernel.

Consequence

Equations of the form T x = y are solvable up to a finite-dimensional obstruction; solvability reduces to finitely many compatibility conditions and a uniquely determined solution modulo ker(T).

Reversal

A non-Fredholm operator typically has an infinite-dimensional kernel or cokernel or a non-closed range; such operators lack a well-defined finite index.

Boundary

This concept applies to bounded linear operators on Banach (or Hilbert) spaces; unbounded operators require additional closedness and domain conditions and are treated separately.

Semantic Tension

Often contrasted with compact operators (which have discrete spectrum accumulating at zero) and with invertible operators; Fredholmness is weaker than invertibility but stronger than mere boundedness.

Synthesis

A Fredholm operator is a bounded linear operator between Banach spaces with finite-dimensional defect spaces and closed range, yielding a stable integer index under compact perturbations.