Elements of Topology and functional analysis

 Elements of Topology and functional analysis

Elements of Topology and functional analysis

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The aim of the book is to provide an easy and informal presentation of the important topics of topology and functional analysis included in the curriculum of Mathematics at the M.Sc. level. The exposition of the subject matter is an easy style in contrast to rather difficult treatment by foreign authors and more suitable to the genius of Pakistan students. The book is selfcontained and no prior exposure to algebraic structures is needed to study this book. Each concept is introduced in a simple manner and has been illustrated and explained by examples.

The topics covered in the book form core courses at the Master's level. General or point set topology forms a basic course for all students of higher Mathematics at all the universities of the world. Most of the concepts of topology are generalizations of the notions of real or complex analysis and of geometry.

Functional analysis synthesizes the important topics of set theory, algebra, geometry and general topology. In functional analysis we are concerned with functions defined on certain sets and study their algebraic and analytic properties, keeping in view the general characteristics of the sets under discussion.

The contents of Chapter One through Three deal with those concept that are used in the rest of the book. Metric spaces are Mathematical systems with a distance function defined on them and have been discussed in Chapter Four. The concepts of convergence of sequences and continuity of functions in metric spaces are discussed in Chapter Five and Six. Fundamental notions of topological spaces have been treated in Chapter Seven. The slight variations in definitions and proofs of concepts of topological spaces as compared to those of the metric spaces have been fully classified.

The distinction between certain properties of points and subsets of a topological space has been explained in Chapter Eight. The separation properties of points and subsets of a topological space are given in detail in this chapter.

Chapter Nine deals with compactness in topological spaces. The distinction between compactness, countable compactness and sequential compactness is explained in this chapter.

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