Spectral theory is one of the main branches of modem functional analysis and its application. It is concerned with certain inverse operators, their general properties and their relations to the original operators. Such inverse operators arise quite naturally in connection with the problem of solving equations (system of linear algebraic equations, differential equations, integral equations).In this dissertation it is proved that the spectrum of bounded self-ad joint linear operators T is real and lies in the interval [m, M], where m and M are the infimum and supermom of inner product of Tx and x taken over all x of norm 1, and eigenvectors corresponding to different eigenvalus are orthogonal.

A Dissertation Submitted in partial fulfillment of the Requirements for the Degree of Master of Science in Mathematics.