Metric Spaces PDF Textbook

Metric Spaces Satish Shirali and Harkrishan L. Vasudeva Metric Spaces With 21 Figures Mathematics Subject Classification (2000): S4E35, 54–02 British Library Cataloguing in Publication Data Shirali Satish Metric spaces 1. Metric spaces I. Title II. Vasudeva, Harkrishan L. 514.3’2 ISBN 1852339225 Library of Congress Control Number: 2005923525 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. ISBN 1-85233-922-5 Springer ScienceþBusiness Media springeronline.com ß Springer-Verlag London Limited 2006 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typeset by SPI Publisher Services, Pondicherry, India Printed and bound in the United States of America 12/3830-543210 Printed on acid-free paper SPIN 11334521 Preface Since the last century, the postulational method and an abstract point of view have played a vital role in the development of modern mathematics. The experience gained from the earlier concrete studies of analysis point to the importance of passage to the limit. The basis of this operation is the notion of distance between any two points of the line or the complex plane. The algebraic properties of underlying sets often play no role in the development of analysis; this situation naturally leads to the study of metric spaces. The abstraction not only simplifies and elucidates mathematical ideas that recur in different guises, but also helps econo- mize the intellectual effort involved in learning them. However, such an abstract approach is likely to overlook the special features of particular mathematical developments, especially those not taken into account while forming the larger picture. Hence, the study of particular mathematical developments is hard to overemphasize. The language in which a large body of ideas and results of functional analysis are expressed is that of metric spaces. The books on functional analysis seem to go over the preliminaries of this topic far too quickly. The present authors attempt to provide a leisurely approach to the theory of metric spaces. In order to ensure that the ideas take root gradually but firmly, a large number of examples and counterexamples follow each definition. Also included are several worked examples and exercises. Applications of the theory are spread out over the entire book. The book treats material concerning metric spaces that is crucial for any ad- vanced level course in analysis. Chapter 0 is devoted to a review and systematisation of properties which we shall generalize or use later in the book. It includes the Cantor construction of real numbers. In Chapter 1, we introduce the basic ideas of metric spaces and Cauchy sequences and discuss the completion of a metric space. The topology of metric spaces, Baire’s category theorem and its applications, including the existence of a continuous, nowhere differentiable function and an explicit example of such a function, are discussed in Chapter 2. Continuous map- pings, uniform convergence of sequences and series of functions, the contraction mapping principle and applications are discussed in Chapter 3. The concepts of connected, locally connected and arcwise connected spaces are explained in Chapter 4. The characterizations of connected subsets of the reals and arcwise connected v vi Preface subsets of the plane are also in Chapter 4. The notion of compactness, together with its equivalent characterisations, is included in Chapter 5. Also contained in this chapter are characterisations of compact subsets of special metric spaces. In Chapter 6, we discuss product metric spaces and provide a proof of Tychonoff ’s theorem. The authors are grateful to Dr. Savita Bhatnagar for reading the final draft of the manuscript and making useful suggestions. While writing the book we benefited from the works listed in the References. The help rendered by the staff of Springer- Verlag London, in particular, Ms. Karen Borthwick and Ms. Helen Desmond, in transforming the manuscript into the present book is gratefully acknowledged. Satish Shirali Harkrishan L. Vasudeva Contents 0. Preliminaries............................................................................. 1 0.1. Sets and Functions............................................................... 1 0.2. Relations........................................................................... 4 0.3. The Real Number System...................................................... 4 0.4. Sequences of Real Numbers................................................... 6 0.5. Limits of Functions and Continuous Functions.......................... 8 0.6. Sequences of Functions......................................................... 9 0.7. Compact Sets..................................................................... 10 0.8. Derivative and Riemann Integral............................................. 11 0.9. Cantor’s Construction.......................................................... 13 0.10. Addition, Multiplication and Order in R................................... 17 0.11. Completeness of R............................................................... 19 1. Basic Concepts........................................................................... 23 1.1. Inequalities........................................................................ 23 1. 2. Metric Spaces..................................................................... 27 1.3. Sequences in Metric Spaces.................................................... 37 1.4. Cauchy Sequences............................................................... 44 1.5. Completion of a Metric Space................................................ 54 1.6. Exercises............................................................................ 58 2. Topology of a Metric Space.......................................................... 64 2.1. Open and Closed Sets........................................................... 64 2.2. Relativisation and Subspaces.................................................. 78 2.3. Countability Axioms and Separability...................................... 82 2.4. Baire’s Category Theorem...................................................... 88 2.5. Exercises............................................................................ 98 vii viii Contents 3. Continuity............................................................................... 103 3.1. Continuous Mappings.......................................................... 103 3.2. Extension Theorems............................................................. 109 3.3. Real and Complex-valued Continuous Functions........................ 112 3.4. Uniform Continuity............................................................. 114 3.5. Homeomorphism, Equivalent Metrics and Isometry.................... 119 3.6. Uniform Convergence of Sequences of Functions........................ 123 3.7. Contraction Mappings and Applications................................... 132 3.8. Exercises............................................................................ 143 4. Connected Spaces..................................................................... 156 4.1. Connectedness.................................................................... 156 4.2. Local Connectedness............................................................ 163 4.3. Arcwise Connectedness......................................................... 165 4.4. Exercises............................................................................ 167 5. Compact Spaces........................................................................ 170 5.1. Bounded sets and Compactness.............................................. 171 5.2. Other Characterisations of Compactness................................... 178 5.3. Continuous Functions on Compact Spaces................................ 182 5.4. Locally Compact Spaces........................................................ 185 5.5. Compact Sets in Special Metric Spaces..................................... 188 5.6. Exercises............................................................................ 194 6. Product Spaces......................................................................... 201 6.1. Finite and Infinite Products of Sets.......................................... 201 6.2. Finite Metric Products.......................................................... 202 6.3. Infinite Metric Products........................................................ 208 6.4. Cantor Set......................................................................... 212 6.5. Exercises............................................................................ 215 Index............................................................................................ 219 0 Preliminaries We shall find it convenient to use logical symbols such as 8, 9, 3, ) and ,. These are listed below with their meanings. A brief summary of set algebra and functions, which will be used throughout this book, is included in this chapter. The words ‘set’, ‘class’, ‘collection’ and ‘family’ are regarded as synonymous and no attempt has been made to define these terms. We shall assume that the reader is familiar with the set R of real numbers as a complete ordered field. However, Section 0.3 is devoted to review and systematisation of the properties that will be needed later, The concepts of convergence of real sequences, limits of real-valued functions, continuity, com- pactness and integration, together with properties that we shall generalise, or that we use later in the book, have been included in Sections 0.4 to 0.8. A sketch of the proof of the Weierstrass approximation theorem for a real-valued continuous function on the closed bounded interval [0,1] constitutes a part of Section 0.8. This has been done for the benefit of readers who may not be familiar with it. The final Sections, Sections 0.9 to 0.11, are devoted to the construction of real numbers from the field Q of rational numbers (axioms for Q are assumed). It is a common sense approach to the study of real numbers, apart from the fact that this construction has a close connection with the completion of a metric space (see Section 1.5). 0.1. Sets and Functions Throughout this book, the following commonly used symbols will be employed: 8 means ‘‘for all’’ or ‘‘for every’’ 9 means ‘‘there exists’’ 3 means ‘‘such that’’ ) means ‘‘implies that’’ or simply ‘‘implies’’ , or ‘‘iff ’’ means ‘‘if and only if ’’. The concept of set plays an important role in every branch of modern math- ematics. Although it is easy and natural to define a set as a collection of objects, it has been shown that this definition leads to a contradiction. The notion of set is, 1 2 0. Preliminaries therefore, left undefined, and a set is described by simply listing its elements or by naming its properties. Thus {x1 , x2 ,... , xn } is the set whose elements are x1 , x2 ,... , xn ; and {x} is the set whose only element is x. If X is the set of all elements x such that some property P(x) is true, we shall write X ¼ {x : P(x)}: The symbol 1 denotes the empty set. We write x 2 X if x is a member of the set X; otherwise, x 62 X. If Y is a subset of X, that is, if x 2 Y implies x 2 X, we write Y X. If Y X and X Y , then X ¼ Y. If Y X and Y 6¼ X, then Y is proper subset of X. Observe that 1 X for every set X. We list below the standard notations for the most important sets of numbers: N the set of all natural numbers Z the set of all integers Q the set of all rational numbers R the set of all real numbers C the set of all complex numbers. Given two sets X and Y, we can form the following new sets from them: X [ Y ¼ {x : x 2 X or x 2 Y }, X \ Y ¼ {x : x 2 X and x 2 Y }: X [ Y and X \ Y are the union an intersection, respectively, of X and Y. If {Xa } is a collection of sets, where a runs through some indexing set L, we write [ \ Xa and Xa a2L a2L for the union and intersection, respectively, of Xa : [ Xa ¼ {x : x 2 Xa for at least one a 2 L}, a2L \ Xa ¼ {x : x 2 Xa for every a 2 L}: a2L If L ¼ N, the set of all natural numbers, the customary notations are [ 1 \ 1 Xn and Xn : n¼1 n¼1 If no two members of {Xa } have any element in common, then {Xa } is said to be a pairwise disjoint collection of sets. If Y X, the complement of Y in X is the set of elements that are in X but not in Y, that is, X\Y ¼ {x : x 2 X, x 62 Y }: 0.1. Sets and Functions 3 The complement of Y is denoted by Y c whenever it is clear from the context with respect to which larger set the complement is taken. If {Xa } is a collection of subsets of X, then De Morgan’s laws hold: !c !c [ \ \ [ Xa ¼ c (Xa ) and Xa ¼ (Xa )c : a2L a2L a2L a2L The Cartesian product X1 X2 ... Xn of the sets X1 , X2 ,... , Xn is the set of all ordered n-tuples (x1 , x2 ,... , xn ), where xi 2 Xi for i ¼ 1, 2,... , n. The symbol f :X !Y means that f is a function (or mapping) from the set X into the set Y ; that is, f assigns to each x 2 X an element f (x) 2 Y. The elements assigned to elements of X by f are often called values of f. If A X and B Y , the image of A and inverse image of B are, respectively, f (A) ¼ {f (x) : x 2 A}, 1 f (B) ¼ {x : f (x) 2 B}: Note that f 1 (B) may be empty even when B 6¼ 1. The domain of f is X and the range is f (X). If f (X) ¼ Y , the function f is said to map X onto Y (or the function is said to be surjective). We write f 1 (y) instead of f 1 ({y}) for every y 2 Y. If f 1 (y) consists of at most one element for each y 2 Y , f is said to be one-to-one (or injective). If f is one-to-one, then f 1 is a function with domain f (X) and range X. A function that is both injective and surjective is said to be bijective. If {Xa : a 2 L} is any family of subsets of X, then ! [ [ f Xa ¼ f (Xa ) a2L a2L and ! \ \ f Xa ¼ f (Xa ): a2L a2L Also, if {Ya : a 2 L} is a family of subsets of Y, then ! [ [ 1 f Ya ¼ f 1 (Ya ) a2L a2L and ! \ \ f 1 Ya ¼ f 1 (Ya ): a2L a2L 4 0. Preliminaries If Y1 and Y2 are subsets of Y, then f 1 (Y1 \Y2 ) ¼ f 1 (Y1 )\ f 1 (Y2 ): Finally, if f : X ! Y and g: Y ! Z, the composite function g f : X ! Z is defined by (g f )(x) ¼ g(f (x)): 0.2. Relations Let X be any set. By a relation R on X, we simply mean a subset of X X. If (x, y) 2 R, then x is said to be in relation R with y and this is denoted by xRy. Among the most interesting relations are the equivalence relations. A relation is said to be an equivalence relation if it satisfies the following three properties: (i) xRx for each x 2 X (reflexive); (ii) if xRy, then yRx (symmetric); (iii) if xRy and yRz, then xRz (transitive). Let R be an equivalence relation on a set X. Then the equivalence class determined by x 2 X is defined by [x] ¼ {y 2 X : xRy}. It is easy to check that any two equivalence classes are either disjoint or else they coincide. Since x 2 [x], it follows thatSR partitions X; that is, there exists a family {Aa : a 2 L} of sets such that X ¼ a2L Aa. Conversely,S if a pairwise disjoint family {Aa : a 2 L} of sets partitions X, that is, X ¼ a2L Aa , then by letting R ¼ {(x, y) 2 X X : 9a 2 L 3 x 2 Aa and y 2 Aa }, an equivalence relation is defined on X whose equivalence classes are precisely Aa. 0.3. The Real Number System We assume that the reader has familiarity with the set R of real numbers and those of its basic properties, which are usually treated in an elementary course in analysis, namely, that it satisfies field axioms, the linear ordering axioms and the least upper bound axiom. In the present section, they are listed in detail. Beginning with the set of natural numbers N, it can be shown that there exists a unique set R that satisfies these properties. The process, though, is lengthy and tedious. Later in the chapter we shall sketch one way of constructing R from Q. A. Field Axioms For all real numbers x, y and z, we have (i) x þ y ¼ y þ x, (ii) (x þ y) þ z ¼ x þ (y þ z), (iii) there exists 0 2 R such that x þ 0 ¼ x, 0.3. The Real Number System 5 (iv) there exists a w 2 R such that x þ w ¼ 0, (v) xy ¼ yx, (vi) (xy)z ¼ x(yz), (vii) there exists 1 2 R such that 1 6¼ 0 and x 1 ¼ x, (viii) if x is different from 0, there exists a w 2 R such that xw ¼ 1, (ix) x(y þ z) ¼ xy þ xz. The second group of properties possessed by the real numbers has to do with the fact that they are ordered. They can be phrased in terms of positivity of real numbers. When we do this, our second group of axioms takes the following form. B. Order Axioms The subset P of positive real numbers satisfies the following: (i) P is closed with respect to addition and multiplication, that is, if x, y 2 P, then so are x þ y and xy, (ii) x 2 P implies x 62 P, (iii) x 2 R implies x ¼ 0 or x 2 P or x 2 P. Any system satisfying the axioms of groups A and B is called an ordered field, for example, the rational numbers. In an ordered field we define the notion x < y to mean y x 2 P. We write x # y to mean x < y or x ¼ y. Absolute value is defined in any ordered field in the familiar manner: x if x $ 0, jxj ¼ x if x < 0: It can be shown on the basis of this definition that the triangle inequality jx þ yj # jxj þ jyj or equivalently, jx yj # jx zj þ jz xj holds. The third group of properties of real numbers contains only one axiom, and it is this axiom that sets apart the real numbers from other ordered fields. Before stating this axiom, we need to define some terms. Let X be a nonempty subset of R. If there exists M such that x # M for all x 2 X, then X is said to be bounded above and M is said to be an upper bound of X. If there exists m such that x $ m for all x 2 X, then X is said to be bounded below and m is said to be a lower bound of X. If X is bounded above as well as below, then it is said to be bounded. A number M 0 is called the least upper bound (or supremum) of X if it is an upper bound and M 0 # M for each upper bound M of X. The final axiom guarantees the existence of least upper bounds for nonempty subsets of R that are bounded above. 6 0. Preliminaries C. Completeness Axiom Every nonempty subset of R that has an upper bound possesses a least upper bound. We shall denote the least upper bound of X by sup X or by sup {x : x 2 X} or by supx2X x. The greatest lower bound or infimum can be defined similarly. It follows from C above that every nonempty subset of R that has a lower bound possesses a greatest lower bound. The greatest lower bound of X is denoted by inf X or by inf {x : x 2 X} or by inf x2X x. Note that inf x2X x ¼ supx2X (x). The following characterisation of supremum is used frequently. Proposition 0.3.1. Let X be a nonempty set of real numbers that is bounded above. Then M ¼ sup X if and only if (i) x # M for all x 2 X, and (ii) given any e > 0, there exists x 2 X such that x > M e. There is a similar characterisation of the infimum of a nonempty set of real numbers that is bounded below. The Cartesian product R R becomes a field under þ and. defined as follows: (x1 , y1 ) þ (x2 , y2 ) ¼ (x1 þ x2 , y1 þ y2 ), (x1 , y1 ) (x2 , y2 ) ¼ (x1 x2 y1 y2 , x1 y2 þ y1 x2 ): It is convenient to denote the ordered pair (x,0) by x and the ordered pair (0, 1) by i. The reader will check that (0, 1) (0, 1) ¼ ( 1, 0), i.e., i 2 ¼ 1. The ordered pair (x, y) can now be written as (x, y) ¼ (x, 0) þ (0, y) ¼ (x, 0) þ (0, 1) (y, 0) ¼ x þ iy: This field is denoted by C and is called the field pof complex ﬃ numbers. ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The absolute value of x þ iy is defined to be x 2 þ y 2. The triangle inequality, as stated above, holds for complex numbers x, y and z. 0.4. Sequences of Real Numbers Functions that have the set N of natural numbers as domain play an important role in analysis. Such functions have special terminology and notation, which we describe below. A sequence of real numbers is a map x : N ! R. Given such a map, we denote x(n) by xn, and this value is called the nth term of the sequence. The sequence itself is frequently denoted by {xn }n $ 1. It is important to distinguish between the sequence {xn }n $ 1 and its range {xn : n 2 N}, which is a subset of R. A real number l is said to be a limit of the sequence {xn }n $ 1 if for each e > 0, there is a positive integer n0 such that for all n $ n0 , we have jxn lj < e. It is easy to verify that a sequence has at most one limit. When {xn }n $ 1 does have a limit, we denote it by lim xn. In symbols, 0.4. Sequences of Real Numbers 7 l ¼ lim xn if 8e > 0, 9n0 3 n $ n0 ) jxn lj < e: A sequence that has a limit is said to converge (or to be convergent). A sequence {xn }n $ 1 of real numbers is said to be increasing if it satisfies the inequalities xn # xnþ1 , n ¼ 1, 2,... ; and decreasing if it satisfies the inequalities xn $ xnþ1 , n ¼ 1, 2,... : We say that the sequence is monotone if it is either increasing or it is decreasing. A sequence {xn }n $ 1 of real numbers is said to be bounded if there exists a real number M > 0 such that jxn j # M for all n 2 N. The following simple criterion for the convergence of a monotone sequence is very useful. Proposition 0.4.1. A monotone sequence of real numbers is convergent if and only if it is bounded. Let {xn }n $ 1 be a sequence of real numbers and let r1 < r2 <... < rn <... be a strictly increasing sequence of natural numbers. Then {xrn }n $ 1 is called a subse- quence of {xn }n $ 1. Proposition 0.4.2. (Bolzano–Weierstrass Theorem) A bounded sequence of real numbers has a convergent subsequence. The convergence criterion described in Proposition 0.4.1 is restricted to mono- tone sequences. It is important to have a condition implying the convergence of a sequence of real numbers that is applicable to a larger class and preferably does not require knowledge of the value of the limit. The Cauchy criterion gives such a condition. A sequence {xn }n $ 1 of real numbers is said to be a Cauchy sequence if, for every e > 0, there exists an integer n0 such that jxn xm j < e whenever n $ n0 and m $ n0. In symbols, 8e > 0, 9n0 3 (m $ n0 , n $ n0 ) ) jxn xm j < e: Proposition 0.4.3. (Cauchy Convergence Criterion) A sequence of real numbers converges if and only if it is a Cauchy sequence. Let {xn }n $ 1 be a bounded sequence in R. Then the limit superior of {xn }n $ 1 is defined by lim sup xn ¼ inf {sup xk }, n k$n and the limit inferior of {xn }n $ 1 is defined by lim inf xn ¼ sup { inf xk }: n k$n Observe that a sequence {xn }n $ 1 is convergent if and only if lim sup xn ¼ lim inf xn : In case {xn }n $ 1 is not bounded from above [respective, below], one defines lim sup xn ¼ 1 [respective, lim inf xn ¼ 1]. 8 0. Preliminaries 0.5. Limits of Functions and Continuous Functions Mathematical analysis is primarily concerned with limit processes. We have already met one of the basic limit processes, namely, convergence of a sequence of real numbers. In this section, we shall recall the notion of the limit of a function, which is used in the study of continuity, differentiation and integration. The notion is parallel to that of the limit of a sequence. We shall also state the definition of continuity and its relation to limits. A point a 2 R is said to be a limit point of a subset X R if every neighbour- hood (a e, a þ e), e > 0, of a contains a point x 6¼ a such that x 2 X. Let f be a real-valued function defined on a subset X of R and a be a limit point of X. We say that f (x) tends to l as x tends to a if, for every e > 0, there exists some d > 0 such that jf (x) lj < e 8x 2 X for which 0 < jx aj < d: The number l is said to be the limit of f(x) as x tends to a, and we write lim f (x) ¼ l or f (x) ! l as x ! a: x!a Note that f (a) need not be defined for the above definition to make sense. Moreover, the value l of the limit is uniquely determined when it exists. The following important formulation of limit of a function is in terms of limits of sequences. Proposition 0.5.1. Let f : X ! R and let a be a limit point of X. Then limx!a f (x) ¼ l if and only if, for every sequence {xn }n $ 1 in X that converges to a and xn 6¼ a for every n, the sequence {f (xn )}n $ 1 converges to l. Let f be a real-valued function whose domain of definition is a set X of real numbers. we say that f is continuous at the point x 2 X if, given e > 0, there exists a d > 0 such that for all y 2 X with jy xj < d, we have jf (y) f (x)j < e. The function is said to be continuous on X if it is continuous at every point of X. If we merely say that a function is ‘continuous’, we mean that it is continuous on its domain. It may checked that f is continuous at a limit point a 2 X if and only if f (a) is defined and limx!a f (x) ¼ f (a). The following criterion of continuity of f at a point a 2 X follows immediately from the preceding criterion and Proposition 0.5.1. Proposition 0.5.2. Let f be a real-valued function defined on a subset X of R and a 2 X be a limit point of X. Then f is continuous at a if and only if, for every sequence {xn }n $ 1 in X that converges to a and xn 6¼ a for every n, lim f (xn ) ¼ f ( lim xn ) ¼ f (a). This result shows that continuous functions are precisely those which send convergent sequences into convergent sequences, in other words, they ‘preserve’ convergence. 0.6. Sequences of Functions 9 The next result, which is known as the Bolzano intermediate value theorem, guarantees that a continuous functions on an interval assumes (at least once) every value that lies between any two of its values. Proposition 0.5.3. Let I be an interval and f : I ! R be a continuous mapping on I. If a, b 2 I and a 2 R satisfies f (a) < a < f (b) or f (a) > a > f (b), then there exists a point c 2 I between a and b such that f (c) ¼ a. 0.6. Sequences of Functions Let X be a subset of R. If to every n ¼ 1, 2,... is assigned a real-valued function fn defined on X, then {fn }n $ 1 is called a sequence of functions on X. The sequence {fn }n $ 1 is said to converge pointwise on X if for each x 2 X, the sequence f1 (x), f2 (x),... of real numbers is convergent. In this case we define a function f on X by taking f (x) ¼ lim fn (x) for every x 2 X. This function is called the pointwise limit of the sequence {fn }n $ 1. Thus a function f which is defined on X is the pointwise limit of the sequence {fn }n $ 1 if, given x 2 X and e > 0, there is an integer n0 (depending on both x and e) such that we have jf (x) fn (x)j < e for all n $ n0. In symbols, given e > 0 and x 2 X, 9 an integer n0 ¼ n0 (x, e) 3 n $ n0 ) jf (x) fn (x)j < e: P1 Recall that a series P n¼1 xn of real numbers converges to x 2 R if the sequence n {sn }n $ 1 , where P1ns ¼ k¼1 xk (the nth partial sum), convergesP to x. We write x ¼ lim sn ¼ n¼1 xn and x is called theP sum of the series. If 1 n¼1 fn (x) converges for every x 2 X, and ifPwe define f (x) ¼ 1 n¼1 fn (x), x 2 X, the function f is called the sum of the series 1 n¼1 f n. A sequence {fn }n $ 1 of functions defined on a set X R is said to converge uniformly on X to a function f if, given e > 0, there is an integer n0 (depending on e only) such that for all x 2 X and all n $ n0 , we have jf (x) fn (x)j < e. In symbols, given e > 0, 9 an integer n0 ¼ n0 (e) 3 n $ n0 ) jf (x) fn (x)j < e 8x 2 X: The statement ‘{fn }n $ 1 converges uniformly to f ’ is written as ‘lim fn ¼ f uniformly’ or as ‘lim fn ¼ f (unif)’. It is clear that every uniformly convergent sequence is pointwise convergent. The converse is, however, notP true. We say that the series 1 n¼1Pfn converges uniformly on X if the sequence {sn }n $ 1 of functions, where sn (x) ¼ nk¼1 fk (x), x 2 X, converges uniformly on X. The Cauchy criterion of uniform convergence of sequences of functions is as follows. Proposition 0.6.1. The sequence of functions {fn }n $ 1 defined on X converges uniformly on X if and only if, given e > 0, there exists an integer n0 such that, for all x 2 X and all n $ n0 , m $ n0 , we have jfn (x) fm (x)j < e. 10 0. Preliminaries The limit of a uniformly convergent sequence of continuous functions is con- tinuous. More precisely, the following is true: Proposition 0.6.2. Suppose {fn }n $ 1 is a sequence of continuous functions defined on X that converges uniformly to f. Then f is continuous. 0.7. Compact Sets The notion of compactness, which is of enormous significance in the study of metric spaces, or more generally in analysis, is an abstraction of an important property possessed by certain subsets of real numbers. The property in question asserts that every open cover of a closed and bounded subset of R has a finite subcover. This simple property of closed and bounded subsets has far reaching implications in analysis; for example, a real-valued continuous function defined on [0,1], say, is bounded and uniformly continuous. In what follows, we shall define the notion of compactness in R and list some of its characterisations. To begin with, we recall the definition of an open subset of R. A subset G of R is said to be open if for each x 2 G, there is a neighbourhood (x e, x þ e), e > 0, of x that is contained in G. Let X be a subset of R. An open cover (covering) of X is a collection C ¼ {Ga : a 2 L} of open sets in R whose union contains X, that is, [ X Ga : a If C 0 is a subcollection of C such that the union of sets in C 0 also contains X, then C 0 is called a subcover (or subcovering) from C of X. If C 0 consists of finitely many sets, then we say that C 0 is a finite subcover (or finite subcovering). A subset X of R is said to be compact if every open cover of X contains a finite subcover. The following proposition characterises compact subsets of R. Proposition 0.7.1. (Heine-Borel Theorem) Let X be a set of real numbers. Then the following statements are equivalent: (i) X is closed and bounded. (ii) X is compact. (iii) Every infinite subset of X has a limit point in X. Proposition 0.7.2. Let f be a real-valued continuous function defined on the closed bounded interval I ¼ [a, b]. Then f is bounded on I and assumes its maximum and minimum values on I, that is, there are points x1 and x2 in I such that f (x1 ) # f (x) # f (x2 ) for all x 2 X. For our next proposition we shall need the following definition. Let f be a real-valued continuous function defined on a set X. Then f is said to be uniformly continuous on X if, given e > 0, there is a d > 0 such that for all x, y 2 X with jx yj < d, we have jf (x) f (y)j < e. 0.8. Derivative and Riemann Integral 11 Proposition 0.7.3. If a real-valued function f is continuous on a closed and bounded interval I, then f is uniformly continuous on I. 0.8. Derivative and Riemann Integral A function f : [a, b] ! R is differentiable at a point c 2 [a, b] if f (c þ h) f (c) lim h!0 h exists, in which case, the limit is called the derivative of f at c and is denoted by f 0 (c). Let [a, b] be a closed and bounded interval of R and f a real-valued function defined on [a, b]. As is well known, the (Riemann) integral ðb f (x)dx a is defined as the limit of Riemann sums X n xj )(xj xj1 ), f (~ j ¼1 where xj , j ¼ 0, 1, 2,... , n, form a partition of [a, b], a ¼ x0 < x1 <... < xn ¼ b, and x~j 2 [xj1 , xj ], j ¼ 1, 2,... , n are arbitrary. Recall that the integral exists, for instance, if f is continuous or monotone. If fÐ and g are Riemann Ðb integrable on [a, b] and f (x) # g(x) for each x 2 [a, b], b then a f (t)dt # a g(t)dt. If f is Riemann integrable Ð b on [a, b], fÐ b$ 0 and [a, b] [a, b], then f is Riemann integrable on [a, b] and a f (t)dt # a f (t)dt. The following is known as the fundamental theorem of integral calculus. Proposition 0.8.1. Let f : [a, b] ! R be integrable and let ðx F(x) ¼ f (t)dt, a < x < b: a Then F is continuous on [a, b]. Moreover, if f is continuous at a point c 2 [a, b], then F is differentiable at c and F 0 (c) ¼ f (c): Proposition 0.8.2. Suppose w has a continuous derivative on [a, b] and f is con- tinuous on the image of the interval [a, b]. Then 12 0. Preliminaries ð w(b) ðb f (t)dt ¼ f (w(u))w0 (u)du: w(a) a Proposition 0.8.3. Suppose {fn }n $ 1 is a sequence of Riemann integrable functions on [a, b] with uniform limit f. Then f is Riemann integrable on [a, b] and ðb ðb lim fn (t)dt ¼ f (t)dt: a a We end this section with a sketch of the proof of the Weierstrass approximation theorem. Proposition 0.8.4. Suppose f is a real-valued continuous function defined on [0, 1]. Then there exists a sequence {Pn }n $ 1 of polynomials with real coefficients that converges uniformly to f on [0,1], that is, 8e > 0, 9n0 3 n $ n0 ) jPn (t) f (t)j < e 8t 2 ½0, 1: Proof. Without loss of generality, we may assume that f (0) ¼ f (1) ¼ 0. This is because g(t) ¼ f (t) f (0) ( f (1) f (0))t is a continuous function satisfying g(0) ¼ g(1) ¼ 0, and if {Qn }n $ 1 is a sequence of polynomials converging uniformly to g, then the sequence of polynomials {Pn }n $ 1 , where Pn (t) ¼ Qn (t) þ ( f (1) f (0) )t þ f (0), converges uniformly to f. Extend the function f to the whole of R by setting f (t) ¼ 0 for t 2 R\[0, 1]. The extended function is clearly continuous on R. For n ¼ 1, 2,... , let an (1 t 2 )n jtj # 1 Qn (t) ¼ 0 jtj > 1, where ð1 1 ¼ (1 t 2 )n dt, an 1 and let ð1 Pn (t) ¼ f (t s)Qn (s)ds: (0:1) 1 Since Qn (t) ¼ Qn ( t) and (1 t 2 )n $ 1 nt 2 for 1 # t # 1, ð1 ð1 ð 1=pﬃﬃn ð 1=pﬃﬃn 4 1 (1 t 2 )n dt ¼ 2 (1 t 2 )n dt $ 2 (1 t 2 )n dt $ 2 (1 nt 2 )dt $ pﬃﬃﬃ $ p : 1 0 0 0 3 n n pﬃﬃﬃ So, an # n. For any d > 0, this implies pﬃﬃﬃ Qn (t) # n(1 d2 )n , where d # jtj # 1: (0:2) 0.9. Cantor’s Construction 13 R1 It is obvious that 1 Qn (t)dt ¼ 1. The function f, being continuous on [0,1], is uniformly continuous; given e > 0, there exists a d > 0 such that s, t 2 [0, 1] and js tj < d imply jf (s) f (t)j < e=2. As f vanishes outside [0, 1], we have jf (s) f (t)j < e=2 for all s, t 2 R with js tj < d. Since f (t) ¼ 0 for t 2 R\[0, 1], it follows by a simple change of variable that ð1 Pn (t) ¼ f (s)Qn (t s)ds: (0:3) 0 Now, the integral on the right of (3) is a polynomial in t. Thus, {Pn }n $ 1 is a sequence of polynomials. Let M ¼ supjf (t)j : t 2 R. For 0 # t # 1 and any positive integer n, using (0.1), (0.2) and (0.3), we have ð1 jPn (t) f (t)j ¼ [f (t s) f (t)]Qn (s)ds 1 ð1 # jf (t s) f (t)jQn (s)ds 1 ð d e ð d ð1 # 2M Qn (s)ds þ Qn (s)ds þ 2M Qn (s)ds 1 2 d d pﬃﬃﬃ e # 4M n(1 d2 )n þ : 2 pﬃﬃﬃ This is less than e for sufficiently large n, because when 0 < d < 1, lim n(1 d2 )n can be shown to be 0 as follows: pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 0 # n(1 d2 )n ¼ n(1 þ b)n # b1 = n, where (1 d2 ) ¼ 1=b, b > 0: & 0.9. Cantor’s Construction In this section we sketch one way of constructing R from Q (the axioms for Q will be assumed). The reasons for the inclusion of this approach are twofold; firstly, it is one of the quickest ways of obtaining R, and secondly, it has a close connection with completion of metric spaces (see Section 1.5). Definition 0.9.1. Let Q denote the field of rational numbers. A sequence {xn }n $ 1 in Q is said to be bounded if there exists a rational number K such that jxn j # K for all n: Definition 0.9.2. A sequence {xn }n $ 1 in Q is said to be Cauchy if for each rational number e > 0 there exists an integer n0 such that jxn xm j < e for all n, m $ n0 : 14 0. Preliminaries Definition 0.9.3. A sequence {xn }n $ 1 in Q is said to converge to a rational number x if for each rational number e > 0 there exists an integer n0 such that jxn xj < e for all n $ n0 : In symbols, limn!1 xn ¼ x. The rational number x is called the limit of the sequence. It may be easily verified that (i) A convergent sequence in Q is a Cauchy sequence in Q; (ii) a Cauchy sequence in Q is bounded; in particular, every convergent sequence in Q is bounded; (iii) if a sequence {xn }n $ 1 converges to x as well as y, then x ¼ y. Thus, the symbol limn!1 xn is unambiguously defined when the sequence {xn }n $ 1 converges. Let FQ denote the set of all Cauchy sequences in Q. Definition 0.9.4. A sequence {xn }n $ 1 in FQ is said to be equivalent to a sequence {yn }n $ 1 in FQ if and only if limn!1 jxn yn j ¼ 0. In symbols, {xn }n $ 1 {yn }n $ 1. The relation defined in FQ is an equivalence relation, as is shown below: (i) Reflexivity: {xn }n $ 1 {xn }n $ 1 , since jxn xn j ¼ 0 for every n, so that limn!1 jxn xn j ¼ 0. (ii) Symmetry: If {xn }n $ 1 {yn }n $ 1 , then limn!1 jxn yn j ¼ 0; but jxn yn j ¼ jyn xn j for every n and therefore limn!1 jyn xn j ¼ 0, so that {yn }n $ 1 {xn }n $ 1. (iii) Transitivity: Suppose {xn }n $ 1 {yn }n $ 1 and {yn }n $ 1 {zn }n $ 1. Then limn!1 jxn yn j ¼ 0 ¼ limn!1 jyn zn j. Since 0 # jxn zn j # jxn yn jþ jyn zn j for all n, it follows that {xn }n $ 1 {zn }n $ 1. Thus, the relation splits FQ into equivalence classes. Any two members of the same equivalence class are equivalent, while no member of an equivalence class is equivalent to a member of any other equivalence class. The equivalence class containing the sequence {xn }n $ 1 will be denoted by [{xn }n $ 1 ] or simply [xn ] for short, i.e., [xn ] ¼ {{yn }n $ 1 2 FQ : {yn }n $ 1 {xn }n $ 1 }: Henceforth we shall abbreviate {xn }n $ 1 as simply {xn } whenever convenient. Proposition 0.9.5. If {xn } 2 FQ then limn!1 xn ¼ x if and only if {xn } {x}, where {x} denotes the constant sequence with each term equal to x. Proof. If {xn } {x}, then by definition of , it follows that limn!1 jxn xj ¼ 0. On the other hand, if limn!1 jxn xj ¼ 0, then {xn } {x}, since the sequence {x} has limit x. & 0.9. Cantor’s Construction 15 Proposition 0.9.6. If {xn } and {yn } are in FQ , then so are the sequences {xn þ yn } and {xn yn }. Proof. Let e > 0 be a rational number. There exist integers n1 and n2 such that e jxn xm j < for all n, m $ n1 2 and e jyn ym j < for all n, m $ n2 : 2 Let n0 $ max {n1 , n2 }. Then for n, m $ n0 we have j(xn þ yn ) (xm þ ym )j # jxn xm j þ jyn ym j < e: So {xn þ yn } is a Cauchy sequence of rational numbers. Since {xn } and {yn } are Cauchy sequences of rational numbers, there exist rational numbers K1 and K2 such that jxn j # K1 and jyn j # K2 for all n: Now, for n, m $ n0, we have jxn yn xm ym j ¼ jyn (xn xm ) þ xm (yn ym )j # jyn j jxn xm j þ jxm j jyn ym j e e e < K2 þ K1 ¼ (K1 þ K2 ) : 2 2 2 Since K1 and K2 are fixed, it follows that {xn yn } is a Cauchy sequence of rational numbers. & Proposition 0.9.7. If {xn }, {yn }, {xn0 } and {yn0 } are in FQ and {xn } {xn0 }, {yn } {yn0 }, then {xn þ yn } {xn0 þ yn0 } and {xn yn } {xn0 yn0 }. Proof. The fact that {xn þ yn } {xn0 þ yn0 } is easy to prove. We proceed to prove the other part. Since every Cauchy sequence is bounded, there exist rational constants K1 and K2 such that jxn j # K1 and yn0 # K2 for all n: Let e > 0 be a given rational number. Since {xn } {xn0 }, and {yn } {yn0 }, there exists n0 such that for n $ n0 we have e e xn x 0 < and yn y 0 < : n n 2K2 2K1 16 0. Preliminaries For n $ n0 we get xn yn x 0 y 0 # xn (yn y 0 ) þ y 0 (xn x 0 ) n n n n n # jxn j yn yn0 þ yn0 xn xn0 e e < þ ¼ e, 2 2 which, in turn, implies limn!1 xn yn xn0 yn0 ¼ 0. & Proposition 0.9.8. If {xn } is a Cauchy sequence in Q that does not have limit 0, then there exists a Cauchy sequence {yn } in Q such that limn!1 jxn yn 1j ¼ 0. Proof. Since {xn } does not have limit 0, there is a positive number a in Q such that for every n 2 N, there exists some k 2 N such that k $ n and jxk j $ a: Since {xn } is a Cauchy sequence, there exists n0 in N such that a jxn xm j < for m, n $ n0 : 2 If jxk0 j $ a, where k0 > n0 , then a a jxn j ¼ jxk0 (xk0 xn )j $ jxk0 j jxk0 xn j > a ¼ 2 2 for all n $ n0. Hence, xn 6¼ 0 for all n $ n0. Let ( 1 if n < n0 , yn ¼ 1 if n $ n0. xn Then {yn } is a sequence in Q. If e is any positive rational number, there exists ne such that a2 e jxm xn j < for all m, n $ ne : 4 Hence, 2 jxm xn j ae 4 jym yn j ¼ < ¼e jxm jjxn j 4 a2 for all m, n $ max {n0 , ne ). Thus {yn } is a Cauchy sequence in Q. Since xn yn ¼ 1 for all n $ n0 , limn!1 jxn yn 1j ¼ 0. & 0.10. Addition, Multiplication and Order in R 17 0.10. Addition, Multiplication and Order in R Definition 0.10.1. A real number is an equivalence class [xn ] with respect to the equivalence relation defined in FQ. Let R denote the set of all real numbers. If j 2 R is [xn ], then j ¼ { {yn }n $ 1 2 FQ : {yn }n $ 1 {xn }n $ 1 }: Definition 0.10.2. If j ¼ [xn ] and h ¼ [yn ] are in R, we define the sum j1h as j þ h ¼ [xn ] þ [yn ] ¼ [xn þ yn ] and the product jh as jh ¼ [xn ] [yn ] ¼ [xn yn ]: Proposition 0.10.3. The operations of addition and multiplication are well defined. With these operations of addition and multiplication, R is a field. Proof. It follows from Proposition 0.9.7 that addition and multiplication are well defined. It can be easily verified that (R, þ,) is a field with [{1}] and [{0}] serving as the multiplicative and additive identities, respectively, and the additive inverse of [xn ] being [xn ]. When [xn ] 6¼ [{0}], it is possible that xn ¼ 0 for some n; conse- quently, the proof of the existence of its multiplicative inverse requires some care: If [xn ] 6¼ [{0}], then {xn } is not equivalent to {0}, so that limn!1 xn 6¼ 0 in Q. Hence, by Proposition 0.9.8 there is a sequence {yn } in Q such that limn!1 xn yn ¼ 1, i.e., {xn yn } {1}. It follows that [yn ] is a multiplicative inverse of [xn ]. & Definition 0.10.4. A sequence {xn } in Q is said to be a positive sequence if there exists a positive rational number a and a positive integer m such that xn > a for all n $ m: The first of the following two facts can now be easily verified by the reader: I. If {xn } and {yn } are positive sequences of rational numbers, then so are {xn þ yn } and {xn yn }: II. If {xn } is a positive sequence of rational numbers and {xn } {xn0 }, then {xn0 } is also a positive sequence of rational numbers. As regards II, there exists a positive rational number a and a positive integer m1 such that 18 0. Preliminaries xn > a for all n $ m1 : Since {xn } {xn0 }, there exists a positive integer m2 such that xn x 0 < a for n $ m2 : n 2 Let m0 ¼ max {m1 , m2 }. For n $ m0, we have a a xn < xn0 < xn þ : 2 2 So, a a xn0 > a ¼ for n $ m0 : 2 2 Definition 0.10.5. A number j 2 R is said to be a positive real number if it contains a positive sequence. The set of all positive real numbers will be denoted by Rþ. We note that by II above, number j 2 R is positive if and only if every sequence belonging to it is positive. Also, Rþ ¼ {j 2 R : j is positive} ¼ {j 2 R : some {xn } 2 j is a positive sequence of rational numbers}: It is clear from I above that Rþ is closed under addition and multiplication, i.e., if j, h 2 Rþ , then so are j þ h and jh. Proposition 0.10.6. If j 2 R, then one and only one of the following must hold: (i) j ¼ 0, (ii) j 2 Rþ , (iii) j 2 Rþ. Definition 0.10.7. For j, h 2 R, we define j > h if j h 2 Rþ. Also, we define the absolute value jjj of j in the usual manner to be 0 if j ¼ 0, j if j 2 Rþ and to be j if j 2 Rþ. With the above definition of order (the relation >), the field R can be shown to be an ordered field in the sense that the following statements are true: Transitivity: j > h > z implies that j > z. Compatibility of order with addition: j > h and z 2 R implies j þ z > h þ z. Compatibility of order with multiplication: j > h and z > 0 implies jz > hz. Proposition 0.10.8. The mapping i : Q ! R defined by i(x) ¼ {x} is an isomorph- ism of Q into R. Moreover, i preserves order and, hence, also absolute values. Proof. If x, y 2 Q, then i(x þ y) ¼ {x þ y} ¼ {x} þ {y} ¼ i(x) þ i(y) R R 0.11. Completeness of R 19 and i(xy) ¼ {xy} ¼ {x}. {y} ¼ i(x). i(y): R R Moreover, x < y in Q if and only if y x > 0 in Q, so that {y x} is a positive sequence in FQ. & Proposition 0.10.9. For any sequence {xn } 2 FQ , we have j[xn ]j ¼ [jxn j]. Proof. We begin by showing that j[xn ]j ¼ 0 if and only if [jxn j] ¼ 0. To this end, j[xn ]j ¼ [{0}] if and only if {xn } {0}, which, in turn, is equivalent to limn!1 xn ¼ 0. Similarly, [jxn j] ¼ [{0}] if and only if limn!1 jxn j ¼ 0. But limn!1 xn ¼ 0 if and only if limn!1 jxn j ¼ 0. Now consider the case when [xn ] > [{0}]. As noted after Definition 0.10.5, {xn } must be a positive sequence, i.e., there exists a positive rational number a and a positive integer m such that xn > a for n $ m (see Definition 0.10.4). By definition of absolute value in Q, it follows that jxn j ¼ xn for n $ m. This implies that {jxn j} {xn }, so that [jxn j] ¼ [xn ]. On the other hand, by Definition 0.10.7, we also have j[xn ]j ¼ [xn ]. Thus, j[xn ]j and [jxn j] are both equal to [xn ]. The case when [xn ] < [{0}] is similar. & 0.11. Completeness of R Convergence in R is defined exactly as in Definition 0.9.3 but with ‘rational’ replaced by ‘real’. We first prove that every Cauchy sequence of rationals converges in R, or more precisely that its image under the order-preserving isomorphism i: Q ! R has a limit in R. Proposition 0.11.1. If {xn } is a Cauchy sequence of rationals and {xn } 2 j, then limn!1 xn ¼ j in R, i.e., limn!1 i(xn ) ¼ j, where i is as in Proposition 0.10.8. Proof. Let e be a positive real number. By Definition 0.10.5, e contains a positive sequence {zn } of rational numbers. Therefore (see Definition 0.10.4) there exists a positive rational number 2a and some positive integer n1 such that zn > 2a for n $ n1. So, zn a > a for n $ n1 and hence {zn a} is a positive rational sequence. It follows that [zn ] > i(a), i.e., e > i(a): (0:3) Since {xn } is a Cauchy sequence, there exists a positive integer n2 such that jxn xm j < 2a for n, m $ n2. Hence a jxn xm j > a for n, m $ n2. Conse- quently, for each n $ n2 , the sequence {jxn xm j}m $ 1 has the property that ½jxn xm j < i(a): (0:4) 20 0. Preliminaries Therefore, n $ n2 implies ji(xn ) jj ¼ ji(xn ) [xm ]j ¼ j[{xn xm }m $ 1 ]j ¼ [jxn xm j] < i(a) < e, using Proposition 0.10.9, (0.4) and (0.3) in that order. Thus, limn!1 xn ¼ j. & In order to avoid cluttered notation, we shall henceforth denote i(x) by x. Thus, for example, in the next Corollary, jj xj means jj i(x)j. Corollary 0.11.2. If j 2 R and e > 0 in R, then there is an x 2 Q such that jj xj < e is in R. Proof. Let {xn } 2 j. Then by Proposition 0.11.1, limn!1 xn ¼ j. Therefore, there exists n0 2 N such that jj xn j < e in R for n $ n0 : In particular, the number x ¼ xn0 has the property that x 2 Q and jj xj < e in R. & Corollary 0.11.3. If j < h in R, there is a z 2 Q such that j < z < h. Proof. j < h implies 2j ¼ j þ j < j þ h < h þ h ¼ 2h and 2 ¼ [{1}] þ [{1}] > [{0}]. It follows that 1=2 > [{0}] in R and hence that j < (j þ h)=2 < h. Let z ¼ (j þ h)=2. If e ¼ min {z j, h z}, then e > 0 and by Corollary 0.11.2, there is a rational number z such that j # z e < z < z þ e # h. & Corollary 0.11.4. R is Archimedean. Proof. For 0 < j < h in R, let x and y be rational numbers (see Corollary 0.11.3) such that 0 < x < j < h < y < j þ h: Since the field Q is Archimedean, there exists a positive integer n such that nx > y. Therefore, nj > nx > y > h: & Theorem 0.11.5. Every Cauchy sequence of real numbers converges in R. Proof. Let {jn }n $ 1 be a Cauchy sequence in R. By Corollary 0.11.2, for each n 2 N, there exists a rational number xn such that jjn xn j < 1=n: 0.11. Completeness of R 21 We shall first show that {xn }n $ 1 is a Cauchy sequence in Q. For e > 0 in R, there exists m1 2 N such that e jjn jm j < for m, n $ m1 : 3 Choose m2 such that 1 e < for all n $ m2 , n 3 so that e jjn xn j < for all n $ m2 : 3 Let m0 ¼ max {m1 , m2 }. Then n, m $ m0 implies jxn xm j ¼ jxn jn þ jn jm þ jm xm j# jxn jn j þ jjn jm j þ jjm xm j < e: Therefore, {xn }n $ 1 is a Cauchy sequence in Q. Now [xn ] is a real number, say j. We shall show that limn!1 jn ¼ j in R. For this purpose, consider any e > 0 in R. Let m1 and m2 be as above. By Proposition 0.11.1, limn!1 i(xn ) ¼ j in R, and therefore there exists m3 2 N such that 2e jj xn j < in R for n $ m3 : 3 Therefore, for n $ m ¼ max {m2 , m3 }, we have jjn jj ¼ jjn xn þ xn jj < jjn xn j þ jxn jj < e: This completes the proof that limn!1 jn ¼ j in R. & Finally, the following result holds: Theorem 0.11.6. Every nonempty subset of the real numbers that is bounded above has a supremum. Proof. Suppose A is a subset containing an element a and having an upper bound b, so that a # b. Since b a $ 0, for any n 2 N, there exists an m 2 N such that m=n $ b a, i.e., a þ m=n $ b 2 A and therefore a þ m=n is an upper bound for A. Hence, for each n 2 N, the set m Bn ¼ {m 2 N : a þ is an upper bound for A} n is nonempty. Being a nonempty set of natural numbers, Bn must have a least element, say mn. Therefore for each n 2 N, mn yn ¼ a þ n is an upper bound of A, i.e., 22 0. Preliminaries x # yn for every n 2 N and every x 2 A, and also, 1 mn 1 xn ¼ yn ¼aþ #x for some x 2 A: n n Hence, xm # yn for all m, n 2 N: It follows that, for any m, n 2 N, we have 1 1 xn xm # ym xm ¼ and xm xn # yn xn ¼ : m n Therefore, 1 1 jxn xm j # max , for all m, n 2 N: n m We use this inequality to argue that {xn } is a Cauchy sequence. Consider any e > 0. Since R is Archimedean, there exists a positive integer n0 > 1=e. For m, n $ n0, it follows from the above inequality that jxn xm j < e. This shows that {xn } is a Cauchy sequence. Since R is complete (see Theorem 0.11.5) {xn } converges; let limn!1 xn ¼ j in R. We next show that j is an upper bound of A. If not, then there exists an x 2 A such that j < x. Since limn!1 xn ¼ j and R is Archimedean, there exists some n 2 N such that xj 1 xj xn j # jxn jj < and < : 2 n 2 Then yn ¼ xn þ 1=n < (x þ j)=2 þ (x j)=2 ¼ x. But this is impossible because x 2 A and yn is an upper bound of A. This contradiction shows that j is an upper bound of A. Finally, we show that j is less than or equal to every upper bound of A. Suppose not. Consider any real number h < j. Let d ¼ j h > 0. Since limn!1 xn ¼ j, therefore there exists n 2 N (corresponding to the positive number d) such that j xn # jj xn j < d ¼ j h and hence h < xn : But, as observed earlier, xn < x for some x 2 A, whence we have h < x for some x 2 A. This implies that h is not an upper bound of A. Thus no real number less than j can be an upper bound of A. In other words, an upper bound of A cannot be less than j and must therefore, by Proposition 0.10.6, be greater than j or equal. & Remark 0.11.7. The above proof makes no explicit reference to real numbers being equivalence classes of Cauchy sequences. A close examination of the argument shows that it works in any ordered field having the Archimedean property and in which every Cauchy sequence converges. 1 Basic Concepts In many branches of mathematics, it is convenient to have available a notion of distance between elements of an abstract set. For example, the proofs of some of the theorems in real analysis or analytic function theory depend only on a few proper- ties of the distance between points and not on the fact that the points are in R or C. When these properties of distance are abstracted, they lead to the concept of a metric space. The notion of distance between points of an abstract set leads naturally to the discussion of convergence of sequences and Cauchy sequences in the set. Unlike the situation of real numbers, where each Cauchy sequence is convergent, there are metric spaces in which Cauchy sequences fail to converge. A metric space in which every Cauchy sequence converges is called a ‘‘complete’’ metric space. This property plays a vital role in analysis when one wishes to make an existence statement. Our objective in this chapter is to define a metric space and list a large number of examples to emphasise the usefulness and the unifying force of the concept. We also define complete metric spaces, give several examples and describe their elementary properties. In Section 1.5, we shall prove that every metric space can be ‘completed’ in an appropriate sense. 1.1. Inequalities The subject of inequalities has applications in every part of mathematics, and the study of metric spaces is no exception. In fact, the definition of a metric space involves an inequality which is a generalisation of the familiar triangle inequality, satisfied by the distance function in R. (jx yj # jx zj þ jz yj for all x, y, z in R or C.) In this section, we establish some inequalities that will be required for confirming that some of the examples we list are indeed metric spaces. Theses examples will be invoked repeatedly. x Proposition 1.1.1. The function f (x) ¼ , x $ 0, is monotonically increasing. 1þx 23 24 1. Basic Concepts Proof. Let y > x $ 0. Then 1 1 1 1 y x < and so 1 >1 , i:e:, > : & 1þy 1þx 1þy 1þx 1þy 1þx Theorem 1.1.2. For any two real numbers x and y, the following inequality holds: jx þ yj jxj jyj # þ : 1 þ jx þ yj 1 þ jxj 1 þ jyj Proof. Let x and y have the same sign. Without loss of generality, we may assume that x $ 0 and y $ 0, and so jx þ yj xþy x y ¼ ¼ þ 1 þ jx þ yj 1 þ x þ y 1 þ x þ y 1 þ x þ y x y jxj jyj # þ ¼ þ : 1 þ x 1 þ y 1 þ jxj 1 þ jyj Suppose x and y have different signs. We may assume that jxj > jyj. Then jx þ yj # jxj. It follows from Proposition 1.1.1 that jx þ yj jxj jxj jyj # # þ : 1 þ jx þ yj 1 þ jxj 1 þ jxj 1 þ jyj This completes the proof. & The next proposition is the well known arithmetic mean-geometric mean inequality, or AM-GM inequality, for short. Proposition 1.1.3. (AM-GM Inequality) If a > 0 and b > 0 and if 0 < l < 1 is fixed, then al b1l # la þ (1 l)b: (1:1) Proof. Since y ¼ ln x, x > 0, is concave, we have ln (la þ (1 l)b) $ l ln a þ (1 l) ln b, i.e., ln al b1l # ln (la þ (1 l)b): As y ¼ exp x is a strictly increasing function, it follows from the above inequality that al b1l # la þ (1 l)b: & Remark. When x $ 0, y $ 0, p > 1 and 1=p þ 1=q ¼ 1, we have 1.1. Inequalities 25 1 1 xy # x p þ y q : p q The inequality is obvious when either x or y is 0. If x 6¼ 0 6¼ y, then it follows from Proposition 1.1.3 upon writing l ¼ 1=p, al ¼ x and b1l ¼ y. Theorem 1.1.4. (Hölder’s Inequality) Let xi $ 0 and yi $ 0 for i ¼ 1, 2,... , n , and suppose that p > 1 and q > 1 are such that 1=p þ 1=q ¼ 1. Then !1=p !1=q Xn Xn p X n q xi yi # xi yi : (1:2) i¼1 i¼1 i¼1 In the special case when p ¼ q ¼ 2, the above inequality reduces to !1=2 !1=2 Xn X n Xn xi yi # 2 xi 2 yi : (1:3) i¼1 i¼1 i¼1 This is known as the Cauchy-Schwarz inequality. Pn p Pn q Proof. We need consider only the case when xi 6¼ 0 6¼ yi. To begin with, we assume that i¼1 i¼1 X n p X n q xi ¼ 1 ¼ yi (1:4) i¼1 i¼1 In this case, the inequality (1.2) reduces to the form X n xi yi # 1 (1:5) i¼1 To obtain (1.5), we put successively x ¼ xi and y ¼ yi for i ¼ 1, 2,... , n in the inequality of the preceding Remark and then add up the inequalities so obtained. The general case can be reduced to the foregoing special case if we take in place of the numbers xi , yi the numbers xi yi xi0 ¼ 0 1=p , yi ¼ n , P p n P q 1=q xi yi i¼1 i¼1 for which the condition (1.4) is satisfied. It follows by what we have proved in the paragraph above that X n xi0 yi0 # 1: i¼1 This is equivalent to (1.2). This completes the proof. & 26 1. Basic Concepts Theorem 1.1.5. (Minkowski’s Inequality) Let xi $ 0 and yi $ 0 for i ¼ 1, 2,... , n and suppose that p $ 1. Then !1=p !1=p !1=p Xn X n p X n p (xi þ yi ) p # xi þ yi : (1:6) i¼1 i¼1 i¼1 Pn If p ¼ 1,p the inequality (1.6) is self-evident. So, assume that p > 1. We write Proof. i ¼ 1 (xi þ yi ) in the form X n X n X n (xi þ yi )p ¼ xi (xi þ yi )p1 þ yi (xi þ yi )p1 : (1:7) i¼1 i¼1 i¼1 Let q > 1 be such that 1=p þ 1=q ¼ 1. Apply Hölder’s inequality to the two sums on the right side of (1.7) and obtain !1=p !1=q X n X n p X n (xi þ yi )p # xi (xi þ yi )(p1)q i¼1 i¼1 i¼1 !1=p !1=q X n p X n þ yi (xi þ yi )(p1)q i¼1 i ¼1 2 !1=p !1=p 3 !1=q X n X n X n ¼4 5 p p xi þ yi (xi þ yi ) p : i ¼1 i ¼1 i ¼1 Pn p 1=q Dividing Pn both sidesp of this inequality by i ¼1 (xi þ yi ) , we obtain (1.6) in the case i¼1 (xi þ yi ) 6¼ 0. In the contrary case, (1.6) is self-evident. & Theorem 1.1.6. (Minkowski’s Inequality for Infinite Sums) Suppose that P1 p $ p1 and let P1 {x n } n $ 1 , {yn } n $ 1 be sequences P1 of nonnegative terms such that n ¼1 xn and n ¼1 y n p are convergent. Then n ¼1 (xn þ yn ) p is convergent. Moreover, !1=p !1=p !1=p X 1 X 1 X 1 (xn þ yn ) p # xnp þ ynp : n¼1 n¼1 n¼1 Proof. For any positive integer m, we have from Theorem 1.1.5, !1=p !1=p !1=p Xm X m X m (xn þ yn ) p # xnp þ ynp n¼1 n¼1 n¼1 !1=p !1=p X 1 X 1 # xnp þ ynp : n¼1 n¼1 1. 2. Metric Spaces 27 P Thus, {( mn ¼1 (xn þ yn ) ) p 1=p }, which is an increasing sequence of nonnegative real numbers, is bounded above by the sum !1=p !1=p X 1 X1 xnp þ p yn : n¼1 n¼1 P1 It follows that n¼1 (xn þ yn )p is convergent and that the desired inequality holds. & Theorem 1.1.7. Let p > 1. For a $ 0 and b $ 0, we have (a þ b)p # 2p1 (ap þ bp ): Proof. If either a or b is 0, the result is obvious. So assume a > 0 and b > 0. Now the function that maps every positive number x into x p is convex when p > 1. So, a þ b p ap þ bp # , 2 2 i.e., (a þ b)p # 2p1 (ap þ bp ): & 1.2. Metric Spaces The notion of function, the concept of limit and the related concept of continuity play an important role in the study of mathematical analysis. The notion of limit can be formulated entirely in terms of distance. For example, a sequence {xn }n $ 1 of real numbers converges to x if and only if for all e > 0 there exists a positive integer n0 such that jxn xj < e whenever n $ n0. A discerning reader will note that the above definition of convergence depends only on the properties of the distance ja bj between pairs a, b of real numbers, and that the algebraic properties of real numbers have no bearing on it, except insofar as they determine properties of the distance such as, ja bj > 0 when a 6¼ b, ja bj ¼ jb aj and ja gj # ja bj þ jb gj: There are many other sets of elements for which ‘‘distance between pairs of elements’’ can be defined, and doing so provides a general setting in which the notions of convergence and continuity can be studied. Such a setting is called a metric space. The approach through metric spaces illuminates many of the concepts of classical analysis and economises the intellectual effort involved in learning them. We begin with the definition of a metric space. Definition 1.2.1. A nonempty set X with a map d : X X ! R is called a metric space if the map d has the following properties: 28 1. Basic Concepts (MS1) d(x, y) $ 0 x, y 2 X; (MS2) d(x, y) ¼ 0 if and only if x ¼ y; (MS3) d(x, y) ¼ d(y, x) x, y 2 X; (MS4) d(x, y) # d(x, z) þ d(z, y) x, y, z 2 X. The map d is called the metric on X or sometimes the distance function on X. The phrase ‘‘(X, d) is a metric space’’ means that d is a metric on the set X. Property (MS4) is often called the triangle inequality. The four properties (MS1)–(MS4) are abstracted from the familiar properties of distance between points in physical space. It is customary to refer to elements of any metric space as points and d(x, y) as the distance between the points x and y. We shall often omit all mention of the metric d and write ‘‘the metric space X’’ instead of ‘‘the metric space (X, d)’’. This abuse of language is unlikely to cause any confusion. Different choices of metrics on the same set X give rise to different metric spaces. In such a situation, careful distinction between them must be maintained. Suppose that (X, d) is a metric space and Y is a nonempty subset of X. The restriction dY of d to Y Y will serve as a metric for Y, as it clearly satisfies the metric space axioms (MS1)–(MS4); so (Y , dY ) is a metric space. By abuse of language, we shall often write (Y, d) instead of (Y , dY ). This metric space is called a subspace of X or of (X, d) and the restriction dY is called the metric induced by d on Y. Examples 1.2.2. (i) The function d: R R ! Rþ defined by d(x, y) ¼ jx yj is a metric on R, the set of real numbers. To prove that d is a metric on R, we need verify only (MS4), as the other axioms are obviously satisfied. For any x, y, z 2 R, d(x, z) ¼ jx zj ¼ j(x y) þ (y z)j # jx yj þ jy zj ¼ d(x, y) þ d(y, z): It is known as the usual or standard metric on R. (ii) Let X ¼ Rn ¼ {x ¼ (x1 , x2 ,... , xn ): xi 2 R, 1 # i # n} be the set of real n-tuples. For x ¼ (x1 , x2 ,... , xn ) and y ¼ (y1 , y2 ,... , yn ) in Rn , define !1=2 X n d(x, y) ¼ (xi yi ) 2 : i¼1 (For n ¼ 2, d(x, y) ¼ ( (x1 y1 )2 þ (x2 y2 )2 )1=2 is the usual distance in the Car- tesian plane.) To verify that d is a metric on Rn , we need only check (MS4), i.e., if z ¼ (z1 , z2 ,... , zn ), we must show that d(x, y) # d(x, z) þ d(z, y). For k ¼ 1, 2,... , n, set ak ¼ xk zk , bk ¼ zk yk : Then !1=2 !1=2 X n X n d(x, z) ¼ ak2 , d(z, y) ¼ bk2 , k¼1 k¼1 and 1. 2. Metric Spaces 29 !1=2 X n d(x, y) ¼ (ak þ bk ) 2 : k¼1 We must show that !1=2 !1=2 !1=2 X n X n X n (ak þ bk ) 2 # ak2 þ bk2 : (*) k¼1 k¼1 k¼1 Squaring both sides of (*), and using the equality (a þ b)2 ¼ a2 þ 2ab þ b2 , we see that (*) is equivalent to !1=2 !1=2 X n X n X n ak bk # ak2 bk2 , k¼1 k¼1 k¼1 which is just the Cauchy-Schwarz inequality (see Theorem 1.1.4). This metric is known as the Euclidean metric on Rn. When n > 1, Rn1 can be regarded as a subset of Rn in the usual way. The metric induced on Rn1 by the Euclidean metric of Rn is the Euclidean metric of Rn1. (iii) Let X ¼ Rn. For x ¼ (x1 , x2 ,... , xn ) and y ¼ (y1 , y2 ,... , yn ) in Rn , define !1=p X n p dp (x, y) ¼ jxi yi j , i¼1 where p $ 1. Note that when p ¼ 2 this agrees with the Euclidean metric. To verify that dp is a metric, we need only check that dp (x, y) # dp (x, z) þ dp (z, y) when z ¼ (z1 , z2 ,... , zn ) 2 Rn. For k ¼ 1, 2,... , n, set ak ¼ xk zk , bk ¼ zk yk. Then !1=p !1=p Xn p X n p dp (x, z) ¼ jak j , dp (z, y) ¼ jbk j k¼1 k¼1 and !1=p X n dp (x, y) ¼ jak þ bk jp : k¼1 We need to show that !1=p !1=p !1=p X n p X n p X n p jak þ bk j # jak j þ jbk j : k¼1 k¼1 k¼1 However, this is just Minkowski’s inequality (see Theorem 1.1.5). 30 1. Basic Concepts Here again, when n > 1, the metric induced by dp on the subset Rn1 is the corresponding metric of the same kind, i.e., !1=p X n1 p dp (x, y) ¼ jxi yi j when x, y 2 Rn1 : i¼1 (iv) Let X ¼ Rn. For x ¼ (x1 , x2 ,... , xn ) and y ¼ (y1 , y2 ,... , yn ) in Rn , define d1 (x, y) ¼ max jxi yi j: 1#i#n Then d1 is a metric. Indeed, for z ¼ (z1 , z2 ,... , zn ) 2 Rn, we have jxi yi j # jxi zi j þ jzi yi j # max jxi zi j þ max jzi yi j, 1#i#n 1#i#n from which it follows that max jxi yi j # max jxi zi j þ max jzi yi j: 1#i#n 1#i#n 1#i#n Remark. Examples (ii), (iii) and (iv) show that we can define more than one distance function, i.e., different metrics in the set of ordered n-tuples. (v) Let X be any nonempty set whatsoever and let 0 if x ¼ y d(x, y) ¼ 1 if x ¼ 6 y. It may be easily verified that d is a metric on X. It is called the discrete metric on X. If Y is a nonempty subset of X, the metric induced on Y by d is the discrete metric on Y. Sequence spaces provide natural extensions of Examples (i), (iii) and (iv). (vi) The space of all bounded sequences. Let X be the set of all bounded sequences of numbers, i.e., all infinite sequences x ¼ (x1 , x2 ,... ) ¼ {xi }i $ 1 such that supi jxi j < 1: If x ¼ {xi }i $ 1 and y ¼ {yi }i $ 1 belong to X, we introduce the distance d(x, y) ¼ supi jxi yi j: It is clear that d is a metric on X. Indeed, if z ¼ {zi }i $ 1 is in X, then 1. 2. Metric Spaces 31 jxi yi j # jxi zi j þ jzi yi j # supi jxi zi j þ supi jzi yi j ¼ d(x, z) þ d(z, y): Therefore, d(x, y) ¼ supi jxi yi j # d(x, z) þ d(z, y): (vii) The space ‘p. Let X be the set of all sequences x ¼ {xi }i $ 1 such that !1=p X1 p jxi j < 1, p $ 1: i¼1 If x ¼ {xi }i $ 1 and y ¼ {yi }i $ 1 belong to X, we introduce the distance !1=p X 1 p d(x, y) ¼ jxi yi j : i¼1 It is a consequence of the Minkowski inequality for infinite sums (Theorem 1.1.6) that d(x, y) 2 R. Evidently, d(x, y) $ 0, d(x, y) ¼ 0 if and only if x ¼ y, and d(x, y) ¼ d(y, x). The triangle inequality follows from the Minkowski inequality for infinite sums. In the special case when p ¼ 2, the space ‘p is called the space of square summable sequences. (viii) The space of bounded functions. Let S be any nonempty set and B(S) denote the set of all real- or complex-valued functions on S, each of which is bounded, i.e., sup jf (x)j < 1: x2S If f and g belong to B(S), there exist M > 0 and N > 0 such that sup j f (x)j # M and sup jg(x)j # N : x2S x2S It follows that sup jf (x) g(x)j < 1. Indeed, x2S jf (x) g(x)j # jf (x)j þ jg(x)j # sup jf (x)j þ sup jg(x)j, x2S x2S and so 0 # sup jf (x) g(x)j # M þ N : x2S Define d(f , g) ¼ sup jf (x) g(x)j, f , g 2 B(S): x2S 32 1. Basic Concepts Evidently, d(f , g) $ 0, d(f , g) ¼ 0 if and only if f (x) ¼ g(x) for all x 2 S and d(f , g) ¼ d(g, f ). It remains to verify the triangle inequality for B(S). By the triangle inequality for R, we have jf (x) g(x)j # jf (x) h(x)j þ jh(x) g(x)j # sup jf (x) h(x)j þ sup jh(x) g(x)j x2S x2S ¼ d(f , h) þ d(h, g) and so d( f , g) # d(f , h) þ d(h, g), for all f , g, h 2 B(S). The metric d is called the uniform metric (or supremum metric). (ix) The space of continuous functions. Let X be the set of all continuous functions defined on [a,b], an interval in R. For f , g 2 X, define d( f , g) ¼ supx2[a, b] j f (x) g(x)j: The measure of distance between the functions f and g is the largest vertical distance between their graphs (see Figure 1.1). Since the difference of two continuous functions is continuous, the composition of two continuous functions is continu- ous, and a continuous function defined on the closed and bounded interval [a,b] is bounded, it follows that d(f , g) 2 R for all f , g 2 X. It may be verified as in Example (viii) that d is a metric on X. The space X with metric d defined as above is denoted by C[a,b]. All that we have said is valid whether all complex-valued continuous functions are taken into consideration or only real-valued ones are. When it is necessary to specify which, we write CC [a, b] or CR [a, b]. Note that C[a, b] B[a, b] and the metric described here is the one induced by the metric in Example (viii) and is also called the uniform metric (or supremum metric). (x) The set of all continuous functions on [a,b] can also be equipped with the metric ðb d(f , g) ¼ jf (x) g(x)jdx: a Figure 1.1 1. 2. Metric Spaces 33 Figure 1.2 The measure of distance between the functions f and g represents the area between their graphs, indicated by shading in Figure 1.2. If f , g 2 C[a, b], then jf gj 2 C[a, b], and the integral defining d(f , g) is finite. It may be easily verified that d is a metric on C[a, b]. We note that the continuity of the functions enters into the verification of the ‘‘only if ’’ part of (MS2). (xi) Let X ¼ R [ {1} [ { 1}. Define f : X ! R by the rule 8 x > < if 1 < x < 1 f (x) ¼ 1 þ jxj > : 1 if x ¼ 1 1 if x ¼ 1. Evidently, f is one-to-one and 1 # f (x) # 1. Define d on X as follows: d(x, y) ¼ jf (x) f (y)j, x, y 2 X: If x ¼ y then f (x) ¼ f (y) and so d(x, y) ¼ 0. On the other hand, if d(x, y) ¼ 0, then jf (x) f (y)j ¼ 0, i.e., f (x) ¼ f (y). Since f is one-to-one, it follows that x ¼ y. That d satisfies (MS3) and (MS4) is a consequence of the properties of the modulus of real numbers. (xii) On X ¼ R [ {1} [ { 1}, define d(x, y) ¼ j tan1 x tan1 yj, x, y 2 X: (Note that tan1 (1) ¼ p=2, tan1 ( 1) ¼ p=2 and that the function tan1 is one-to-one.) It may now be verified that X is a metric space with metric d defined as above. (xiii) The extended complex plane. Let X ¼ C [ {1}. We represent X as the unit sphere in R3 , S ¼ {(x1 , x2 , x3 ): x1 , x2 , x3 2 R, x12 þ x22 þ x32 ¼ 1}, and identify C with {(x1 , x2 , 0): x1 , x2 2 R}. The line joining the north pole (0,0,1) of S with the point z ¼ (x1 , x2 ) 2 C intersects the sphere S at (j, h, z) 6¼ (0, 0, 1), say. We map C to S by sending z ¼ (x1 , x2 ) to (j, h, z). The mapping is clearly one-to- one and each point of S other than (0,0,1) is the image of some point in C. The correspondence is completed by letting the point 1 correspond to (0, 0, 1). (See Figure 1.3.) 34 1. Basic Concepts Figure 1.3 Analytically, the above representation is described by the formulae 2 2jz1 z2 j >

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