Mohammed, Sied (PhD)Hailu, Abraham2020-12-152023-11-042020-12-152023-11-042010-06-06http://etd.aau.edu.et/handle/123456789/24065In certain problems that we encounter it is common to get equations in which the unknown function occurs under an integral sign. Such equations are termed as integral equations. Integral equations are use full to solve some classes of differential equations. We will see that the solution of a differential equation is an integral equation. This is one nice property that makes integral equations to be studied. The study of integral equations leads to the study of compact operators in a more general setting of infinite dimensional Hilbert spaces. Compact operators in infinite dimensional Hilbert space can be considered as counter parts of matrices as they enjoy many properties in common with linear operators in finite dimensional spaces (matrices). For instance, they satisfy Fredholm alternative theorem and exhibit some spectral properties. The objective of this seminar is therefore to generalize some of the fundamental properties of matrices in finite dimensional case to compact linear operators in infinite dimensional Hilbert spaces. The seminar deals with integral equations in a Hilbert space with emphasis on spectral theorem of eigenvalues and eigenfunctions of a compact operator in an infinite dimensional Hilbert space. Most of the examples and illustrations will be given on space. This seminar also deals with the Fredholm alternative theorem of a general compact linear operator. In this seminar there are four sections. In section one some preliminary concepts that are needed later will be dealt. In section two bounded linear operators in Hilbert space will be studied. Section three deals with compact linear operators and Hilbert-Schmidt kernels. Finaly, Spectral theory on compact operators will be investigated in section four.enIntegral EquationsBasic Preliminary ConceptsThesis