Dual Systems and Fredholm Alternative

Abstract

In the case when the homogeneous equation has nontrivial solutions, the Riesz theory, i.e., Theorem 3.4 gives no answer to the question of whether the inhomogeneous equation for a given inhomogeneity is solvable. This question is settled by the Fredholm alternative, which we shall develop in this chapter. Rather than presenting it in the context of the Riesz–Schauder theory with the adjoint operator in the dual space we will consider the Fredholm theory for compact adjoint operators in dual systems generated by non-degenerate bilinear or sesquilinear forms. This symmetric version is better suited for applications to integral equations and contains the Riesz–Schauder theory as a special case.