ABSTRACT
In this chapter, the authors present parts of the theory of integral equations that are needed for a unified study of Sturm-Liouville problems. Fredholm's approach to multiplicity relies on non-elementary results of complex analysis. Subsequently Issai Schur observed that it would be desirable to give an elementary, but equivalent, formulation of Fredholm's multiplicities that is tied more closely to corresponding matrix results. Many eigenvalue problems that arise in applications are equivalent to eigenvalue problems for integral operators that are self-adjoint. The eigenvalues for self-adjoint matrices are real and there is a corresponding set of orthonormal eigenvectors that are a basis for the underlying real or complex Euclidean space. An application of the Hilbert-Schmidt theorem will prepare the way for Mercer's theorem. Two main results from the theory of integral equations play a pivotal role in a unified study of Sturm-Liouville boundary value and eigenvalue problems.
