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SMA301 LATEST NOTES.Pdf September 15, 2011 UNIVERSITY OF NAIROBI FACULTY OF SCIENCE THIRD YEAR LECTURE NOTES SMA 301: REAL ANALYSIS I First Edition WRITTEN BY : Dr. Bernard Mutuku Nzimbi REVIEWED BY: School of Mathematics, University of Nairobi P.o Box 30197, Nairobi, KENYA. EDITED BY: Copyright ⃝c 2011 Benz, Inc. All rights reserved. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopy- ing, recording or otherwise, without the prior permission of the publisher. This page is intentionally left blank. i Contents About the Author v Preface vi Goals of a Real Analysis I Course viii 1 METRIC SPACES 1 1.1 Introduction . 1 1.2 Rudiments on Metric Spaces . 1 1.3 Examples of Metric and Metric spaces . 3 1.3.1 Useful Inequalities . 6 1.3.2 H¨older's Inequality . 7 1.3.3 Minkowski's Inequality . 8 1.3.4 Subspaces of a metric space . 12 1.4 Equivalent Metrics . 12 1.5 Exercises . 18 2 STRUCTURE OF METRIC SPACES 22 2.1 Topology of a Metric Space . 22 2.1.1 Open Balls and Closed Balls in a Metric Space . 22 2.2 Interior points, Isolated Points and Open sets in a Metric Space . 28 2.2.1 Examples of Open Sets . 30 2.3 Limit Points, Closure Points and Closed Sets in a Metric Space . 33 2.3.1 Examples of Closed Sets . 37 2.4 Subsets and Relative Topology . 40 ii 2.4.1 Relatively Open and Relatively Closed Subsets of Metric Spaces . 41 2.5 Diameter of Subsets and Distance Between Subsets in Metric Spaces . 42 2.6 Boundary of a Subset of a Metric Space . 45 2.7 Exercises . 48 3 DENSE SUBSETS IN A METRIC SPACE 54 3.1 Dense and Nowhere Dense Sets . 54 3.2 Exercises . 56 4 COMPACTNESS IN METRIC SPACES 58 4.1 Sequential Compactness, Notion of a Cover and a Compact Set . 58 4.2 Bolzano-Weierstrass Theorem and Compactness . 59 4.3 Heine- Borel Theorem and Compactness . 62 4.4 Exercises . 64 5 CONTINUITY IN METRIC SPACES 66 5.1 Continuity of a Function in a metric Space . 66 5.1.1 Epsilon-Delta Definition of Continuity in a Metric Space . 66 5.1.2 Continuity in terms of Open Balls . 67 5.2 Compactness and Continuity in Metric Spaces . 70 5.2.1 Uniform Continuity in Metric Spaces . 73 5.2.2 Continuity vs Metrics . 74 5.2.3 Compactness and Real-valued Continuous Functions: Applica- tions in Maxima and Minima Problems . 77 5.2.4 Continuity and Approximation . 78 5.3 Exercises . 79 6 COMPLETENESS IN METRIC SPACES 81 6.1 Convergence in Metric Spaces . 81 6.1.1 Criterion of Convergence . 81 6.1.2 Cauchy Criterion and Cauchy Sequences . 82 6.2 Compactness vs Completeness in Metric Spaces . 86 6.3 Baire Category Theorem . 90 iii 6.4 Completeness vs Equivalent metrics in Metric Spaces . 94 6.5 Exercises . 95 Bibliography 96 iv About the Author B.M. Nzimbi is a Lecturer in Pure Mathematics at the University of Nairobi. Dr. Nzimbi received his B.sc in Mathematics and Computer Science from the University of Nairobi (1995), Msc(Pure Mathematics) from University of Nairobi (1999), Msc(Mathematics) from Syracuse University (New York, USA) (2004), and his Ph.D in Pure Mathematics from University of Nairobi (2009), where he wrote his thesis in the area of Operator Theory under the direction of Prof. J.M. Khalagai. Before joining the University of Nairobi, he held a position at Catholic University of Eastern Africa (CUEA), where he was a part-time lecturer. Dr. Nzimbi has authored, co-authored and published numerous articles in professional journals in the areas of Operator Theory and differential geometry. He is the author of the textbooks "Basic Mathematics","Linear Algebra I " and "Linear Algebra II ", which are extensively used in the ODL programme at the University of Nairobi. v Preface This book grew out of a course entitled Real Analysis I that I have taught at the Uni- versity of Nairobi during the past six years. It is intended as a textbook to be studied independently by students on their own or to be used in a course on Real Analysis I. In writing this book, I was guided by my long-standing experience and interest in teach- ing Real Analysis. The choice of material is not entirely mine, but laid down by the University of Nairobi SMA301: Real Analysis syllabus. For the student, my purpose was to introduce students studying sciences and social sci- ences the beautiful field of real analysis. I wanted to write a succinct book that gets to interesting results in a minimal amount of time. This book can be read by under- graduate students who have had only one-semester course in introduction to analysis. Introduction to Analysis is the only explicit prerequisite for the course, but a cer- tain degree of mathematical sophistication is required. For the instructor, my purpose was to design a flexible, comprehensive teaching tool using proven pedagogical techniques in mathematics. I wanted to provide instructors with a package of materials that they could use to teach Real Analysis I effectively and efficiently in the most appropriate manner for their particular set of students. I hope I have accomplished these goals without watering down the material. This text is designed as a one-semester real analysis course to be taken by third year undergraduate students in a wide variety of majors, including mathematics, chemistry, physics, meteorology, geology, economics and other social sciences. The core material of this book consists of six chapters. Each chapter includes definitions, theorems and principles together with illustrative and other descriptive material, followed by several exercises of varying difficulty. I finally wish to record my appreciation to my former students for their invaluable suggestions and critical review of the manuscript that made the writing of this book easy. I remain indebted to my former students for their patience and eagerness to read my sometimes not-so-polished notes and for much help in sorting out good exercise sets vi which have been included in this book. I welcome your comments, suggestions for improvements and indication of errors. All the errors in this manuscript, if any, are entirely mine. Bernard Mutuku Nzimbi Nairobi, 2011 vii Goals of a Real Analysis I Course Analysis is the branch of mathematics that deals with inequalities and limiting processes. The goal of this course is to acquaint the reader with the basic concepts of rigorous proof in analysis. Special Features of this Book Accessibility: There are no mathematical prerequisites beyond an introduction to analysis for this text. Each chapter begins at an easily understood and accessible level. Once basic mathematical concepts have been developed, more difficult material and ap- plications are presented. Accessibility: This text has been carefully designed for flexible use. The depen- dence of chapters has been minimized. Each chapter is divided into sections and each section is divided into subsections that form natural blocks of material for teaching. Instructors can easily pace their lectures using these blocks. Writing Style: The writing style of this book is direct and pragmatic. Precise mathematical language is used without excessive formalism and abstraction. Notations are introduced and used when appropriate. Care has been taken to balance the mix of notation and words in mathematical statements. Mathematical Rigor and Precision: All definitions and theorems in this book are stated extremely carefully so that students will appreciate the precision of language and rigor need in mathematics. Proofs are motivated and developed and their steps are carefully justified. Figures and Tables: Figures and tables in this book are carefully presented and illustrated. Exercises: There is an ample supply of exercises in this book that develop basic skills and which are carefully graded for level of difficulty. viii Chapter 1 METRIC SPACES 1.1 Introduction One of the most important operations in mathematical analysis is the taking of limits. Here what matters is not so much the algebraic nature of the field of real numbers, but rather the fact that distance from one point to another on the real line (or in two or three dimensional space) is well-defined and has certain properties. All fundamental tools of metric spaces needed in mathematics and elsewhere in the sci- ences and social sciences are included in detailed exposition in this chapter. Objectives At the end of this lecture, you should be able to: • Define a metric and a metric space. • Give examples of metric spaces. • Apply metric space theory in solving some practical problems. 1.2 Rudiments on Metric Spaces Basic questions of analysis on the real line are tied to the notions of closeness and distances between points. The same issue of closeness comes up in more complicated settings, for instance, like when we are trying to approximate a function by a simpler 1 function. The notion of a metric space generalizes the properties of R that are associated with the distance given by the function (x; y) 7−! jx − yj which is the standard metric or Euclidean metric in R. We introduce main properties of matric spaces (applicable in advanced real analysis). We abstract many of the properties of R to the context of a metric space, a set in which we can measure the distance between any two points. We introduce the general notion of distance for the general space emphasizing main properties of the distance with which we have an experience. Once a metric distance has been defined between any two points of a set, the set becomes well defined or metrized and it becomes a space.
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