Fredholm alternative

The Fredholm alternative characterizes solvability of linear equations involving the identity plus a compact operator: either the operator is invertible, or its kernel and adjoint kernel impose compatibility conditions.

Compact operator: A linear operator that maps bounded sets to sets with compact closure. Compactness is the hypothesis that makes identity-perturbed operators obey the alternative.
Rank–nullity theorem: A theorem equating a linear map's domain dimension with its rank plus its nullity. It gives the finite-dimensional dimension balance underlying the solvability alternative.
Ivar Fredholm: A Swedish mathematician whose work on integral equations established foundational results in operator theory. His name is attached to the alternative developed through his study of integral equations.
Boundary value problem: A differential equation problem whose solution must satisfy specified conditions at a domain's boundary. Linear boundary-value problems often reduce to operator equations governed by the alternative.
Banach–Steinhaus theorem: A theorem stating that pointwise bounded families of bounded linear operators are uniformly bounded. Unlike the alternative, it controls families of operators rather than solvability of one equation.
Kernel (linear algebra): The set of vectors a linear map sends to zero. A nontrivial kernel marks the non-invertible branch and causes nonuniqueness.
Fredholm integral equation: An integral equation whose kernel defines a compact integral operator under common conditions. These equations are a principal setting where the theorem turns into a concrete solvability test.
Erik Ivar Fredholm's 1903 paper: Fredholm's paper that developed a theory of linear integral equations and their solvability. It presented the integral-equation framework from which the alternative emerged.
Green's function: A kernel representing the response of a linear differential operator to a point source. It can transform differential equations into integral equations where solvability constraints become visible.
Lax–Milgram theorem: A theorem guaranteeing a unique variational solution from a bounded coercive bilinear form. It gives uniqueness through coercivity, while the Fredholm alternative handles compact perturbations and compatibility.

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