A THESIS SUBMITTED TOTHE DEPARTMENT OF MATHEMATICSCOLLEGE OF NATURAL SCIENCESSCHOOL OF GRADUATE STUDIES ARBAMINCH UNIVERSITYIN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE IN MATHEMATICS WITH SPECIALIZATION IN FUNCTIONAL ANALYSIS

Abstract

Among the Fundamental theorem of Functional Analysis, we study that the “Versatile Hahn – Banach Theorem on the context of real linear vector space and on normed space (Banach Spaces) in its two principal forms which are the continuous extension theorem ( Analytic version) and the separation theorem (Geometric version). We began by discussing some preliminaries that helps to formulate the statement of the theorems along with some definitions that are required to understand these statements and we also prove some of these theorems and discuss some of its important properties, consequences, and its applications. The whole thesis out lines that these theorems are applied in many disciplines, we make some comments that give condition for the theorems as a power full principle of functional analysis.