Concise Introduction to Basic Real Analysis

Hemen Dutta, P. N. Natarajan, Yeol Je Cho

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9780429876332
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This book provides an introduction to basic topics in Real Analysis and makes the subject easily understandable to all learners. The book is useful for those that are involved with Real Analysis in disciplines such as mathematics, engineering, technology, and other physical sciences. It provides a good balance while dealing with the basic and essential topics that enable the reader to learn the more advanced topics easily. It includes many examples and end of chapter exercises including hints for solutions in several critical cases. The book is ideal for students, instructors, as well as those doing research in areas requiring a basic knowledge of Real Analysis. Those more advanced in the field will also find the book useful to refresh their knowledge of the topic.

Features

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• Includes basic and essential topics of real analysis
• Adopts a reasonable approach to make the subject easier to learn
• Contains many solved examples and exercise at the end of each chapter
• Presents a quick review of the fundamentals of set theory
• Covers the real number system
• Discusses the basic concepts of metric spaces and complete metric spaces

1

Review of Set Theory

For the study of any branch of Mathematics, set theory is always very useful. Set theory was developed by Book and Cantor in the latter part of the 19th century. Development of Mathematics in the 20th century was very greatly influenced by set theory. In the present chapter, we introduce some basic concepts of set theory which are very useful in the sequel.

1.1 Introduction and Notations

We do not attempt to define a “set”. All of us agree to consider a “set” as undefined. A set can be “described” as a collection of objects considered as a single entity. The objects of this collection are called elements or members of the set. These objects are said to belong to the set or are contained in the set. The set is said to contain its objects or is composed of its objects.

Sets are usually denoted by capital letters A, B, C, . . . or X, Y, Z, . . ., and elements of the set are denoted by lower-case letters a, b, c, . . . or x, y, z, . . .. If x is an element of a set X, we write x ∈ X. In such a case, we also say that “x belongs to X”. If x does not belong to X, we write x ∈ X. Often, we denote sets by displaying their elements within braces. For example, the set of all positive integers less than 12 is denoted by {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 } or by {x : x is an integer, 0 < x <12 }. More generally, if X is the set of all elements x satisfying a property P , we denoted it by

X={x:xsatifiesP}.

Given a set, we can form new sets as follows. For instance, let Y be the set of all positive integers less than 12 which are divisible by 3; then, Y = { 3, 6, 9 }. Note that Y contains some elements of the set X = {x : x is an integer, 0 < x < 12 }. Y is called a subset of X. More generally, given the sets A, B, we say that A is a subset of B, written as A ⊆ B (or B ⊇ A) if every...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Authors
• 1 Review of Set Theory
• 2 The Real Number System
• 3 Sequences and Series of Real Numbers
• 4 Metric Spaces – Basic Concepts, Complete Metric Spaces
• 5 Limits and Continuity
• 6 Connectedness and Compactness
• 7 Differentiation
• 8 Integration
• 9 Sequences and Series of Functions
• Bibliography
• Index