An Introduction to Modern Analysis

Transcription

1 An Introduction to Modern Analysis

2 Vicente Montesinos Peter Zizler Václav Zizler An Introduction to Modern Analysis 2123

3 Vicente Montesinos Departamento de Matemática Aplicada Instituto de Matemática Pura y Aplicada Universitat Politècnica de València Valencia, Spain Peter Zizler Department of Mathematics, Physics and Engineering Mount Royal University Calgary, Alberta Canada Václav Zizler Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta Canada ISBN ISBN (ebook) DOI / Springer Cham Heidelberg NewYork Dordrecht London Library of Congress Control Number: Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 Preface Mathematics is the queen of the sciences Carl Friedrich Gauss This text is directed at undergraduate students in mathematical sciences who wish to have solid foundations for modern analysis, a meeting point of classical analysis with other parts of mathematics, like functional analysis, operator theory, nonlinear analysis, etc. These foundations are necessary for applications of mathematics in sciences or engineering. Moreover, students planning to pursue graduate work in mathematics will find this text useful, especially those who did not have a chance to go through the honors programs at their respective universities or colleges. It is assumed the reader has a good understanding of elementary linear algebra and arithmetics, as well as some training in simple logic. We shall try to fill foreseeable gaps to help the reader in this direction. The text consists of a rigorous yet gentle self-contained introduction to real analysis with various visual supplements. Moreover, we have enriched the material with several excursions to mathematical areas such as functional analysis, descriptive statistics, or Fourier analysis (some chapters that are rather self-contained can be used as a material for independent optional course in some undergraduate programs). Aside from the theoretical part, the text contains an ample amount of exercises of various difficulties with hints for their solutions. We have prepared a number of figures (by using the free-distribution programs Veusz and IPE, and in a few opportunities also the registered package Mathematica) that are intended to help with understanding of the material covered. We tried to touch on quite a few folklore things that are frequently used in real analysis. We hope that instructors in service calculus courses may find the text to be a source for more advanced problems. In the first chapter we introduce the real number system, discuss the principle of the supremum, and first meet the important principle of compactness and the Baire Category theorem. In the second chapter we encounter the notion of convergent and Cauchy sequences of real numbers and the approximation by rational numbers. v

5 vi Preface Chapter 3 contains an introduction to Lebesgue measure on the real line and its applications. Chapter 4 contains basic notions and results in the theory of real-valued functions and their differentiability, together with an introduction to sequences of real-valued functions and their convergence. Chapter 5 upgrades the discussion on function convergence. We discuss pointwise, uniform, measure, and almost everywhere convergences. The focus is on approximation and the properties preserved through it. In particular, global and local approximations are considered. Applications of those concepts include a discussion on real analytic functions and rigorous definitions of the basic functions in analysis. Chapter 6 deals with metric spaces. This is a wide setting in which most of the former discussions find their place. The reader may find here Tietze s extension theorem, a discussion on separable spaces, with an emphasis on Polish spaces, a deeper analysis of compactness, including the Arzelà Ascoli theorem, more on the Baire category theorem, and applications to metric fixed point theory. Chapter 7 deals with integration in the Riemann and Lebesgue senses. Lebesgue s approach is intertwined with the measure theory already developed, and allows for a finer analysis of functions and convergence. Chapter 8 introduces the reader to the basic theory of convex functions. Chapter 9 is a basic introduction to the theory of Fourier series and integrals, including applications. An extension to the more general setting of periodic distributions will be done in Chap. 11. Chapter 10 presents a basic introduction to descriptive statistics. The emphasis is on discrete probability, which may help to understand the subsequent, more general approach. In Chap. 11, named Excursion to Functional Analysis, we present an introduction to basic concepts and results in a few selected topics in functional analysis, like Banach spaces, operator theory, and nonlinear functional analysis, with applications to real analysis. In fact, we shall try to illustrate to some extent how abstract functional analysis emerges from the waters of real analysis as a lighthouse to orientate and overlook the whole sea. We believe that this chapter may be used as a basic introduction to these subjects, and may foster the interest of the reader to enlarge his/her knowledge of modern techniques used in many fields. Together with Chap. 6, this chapter may constitute a basic material for an introductory graduate course in linear and nonlinear functional analysis. We include an Appendix (Chap. 12), mainly on number systems, and on three fundamental principles in set theory the axiom of choice, the well-ordering principle, and Zorn s lemma. The last chapter (Chap. 13) is formed by exercises that are organized according to the chapters in the text. They are of various levels of difficulty. Some of them just briefly review the basic techniques of rigorous elementary calculus, and some of them upgrade the material in the chapters of the text. All of them are accompanied by hints for their solutions. Optional sections are denoted by the symbol.

6 Acknowledgements The authors thank the institutions where they got the opportunity to teach the material in the text, namely, the University of Alberta (Edmonton, Alberta, Canada), Mount Royal University (Calgary, Alberta, Canada), and The Universitat Politècnica de València (València, Spain). They thank their colleagues and students for many discussions that helped in preparing this text. In particular, we thank M. Fabian, A. J. Guirao, P. Hájek, J. Muldowney and the late M. Valdivia. The authors would like to thank the Springer team for their interest in this text. In particular, they are thankful to Keith F. Taylor, Vaishali Damle, and Marc Strauss. They also thank Sakshi Narang for the assistance and the very professional work done in editing the final version of this book. Above all, the authors are indebted to their families for their moral support and encouragement. They wish the reader a pleasant journey through this book. vii

7 Notation Special symbols used in this book will be introduced along the text. In order to keep track of them, a list of symbols referring to the page where they first appear or where they are defined is included. The first appearance is written in boldface. We tried to follow the usual notation regarding mathematical symbols. However, we depart from this habit in some particular cases. For example, B[x, r] denotes the closed ball with center x and radius r in a metric space. When a symbol for a generic function is needed, we use f, g, or similar, and we speak of the function f. Coherence may force then to speak about the function sin, or the function ln, for example. However, it is a tradition to refer to these functions as to the function sin x, or the function ln x, and we follow this convention. In this text, two notions of integral are used: The Riemann integral and the Lebesgue integral. For a function f defined in an interval [a, b], the first one is denoted by b a f or b a f (x)dx, while the symbol [a,b] f or [a,b] f (x)dx is reserved for the second. Accordingly, if S is a measurable set and f : S R is a Lebesgue integrable function on S, the Lebesgue integral of f on S will be denoted by S f or S f (x)dx. Improper Riemann integrals will be denoted then by + a f or + a f (x)dx. Every Riemann-integrable function f on a closed and bounded interval [a, b] is Lebesgue-integrable, and both integrals coincide. In this case, the common value of the Riemann and the Lebesgue integral will be denoted by b a f (or b a f (x)dx), what seems an accepted practice. The end of a proof is marked, the end of an example, while the end of a remark uses the symbol. ix

8 Contents 1 Real Numbers: The Basics Notation Natural Numbers Integers Fractions and Rational Numbers Introduction Powers and Radicals of Rational Numbers Base Representation The Expansion of a Natural Number in Base b The Expansion of a Rational Number in Base b Real Numbers The Definition of a Real Number The Expansion of a Real Number in Base b The Extended Real Number System, Intervals Order Properties and the Completeness of R Cardinality of Sets Basics on Cardinality Cardinality of Z and Q Cardinality of R Cardinality of the Set of Real Functions Topology of R Introduction. Open and Closed Sets Neighborhoods, Closure, Interior Topology on a Subset Compactness Connectedness and Related Concepts The Baire Category Theorem in R Sequences and Series Approximation by Rational Numbers Sequences Basics on Sequences xi

9 xii Contents Two Particular Sequences: Arithmetic and Geometric Progressions More on Sequences Series Introduction General Criteria for Convergence of Series Series of Nonnegative Terms Series of Arbitrary Terms Rearrangement of Series Double Sequences and Double Series Product of Series The Euler Number e Infinite Products Measure Measure The Lebesgue Outer Measure The Class of Lebesgue Measurable Sets and the Lebesgue Measure Approximating Measurable Sets The Lebesgue Inner Measure The Cantor Ternary Set A Nonmeasurable Set Sequences of Sets Functions Functions on Real Numbers Introduction The Limit of a Function Continuous Functions Differentiable Functions Optimization and the Mean Value Theorem Algebra of Derivatives The Trigonometric Functions Finer Analysis of Continuity and Differentiability Differentiability of the Inverse Mapping Inverse Goniometric Functions Monotone Functions Measurable Functions Differentiability of Monotone Functions Functions of Bounded Variation Absolutely Continuous Functions and Lipschitz Functions Examples The Intermediate Value Property II

10 Contents xiii 5 Function Convergence Function Sequences Pointwise and Almost Everywhere Convergence Uniform Convergence Convergence in Measure Local Approximation by Polynomials Function Series Power Series The Taylor Series The Exponential and the Logarithmic Functions The Hyperbolic Functions The Trigonometric Functions The Binomial Series Metric Spaces Basics Mappings Between Metric Spaces More Examples (Continued) Tietze s Extension Theorem Complete Metric Spaces and the Completion of a Metric Space Separable Metric Spaces Polish Spaces Compactness in Metric Spaces Compact Metric Spaces Total Boundedness Continuous Mappings on Compact Spaces The Lebesgue Number of a Covering The Finite Intersection Property. Pseudocompactness The Baire Category Theorem Continued The Baire Category Theorem in the Context of Metric Spaces Some Applications of the Baire Category Theorem The Arzelà Ascoli Theorem Metric Fixed Point Theory The Banach Contraction Principle Continuity of the Fixed Point Integration The Riemann Integral Introduction The Definition of the Riemann Integral Properties of the Integral Functions Defined by Integrals

11 xiv Contents Some Applications of the Riemann Integral and the Arzelà Ascoli Theorem to the Theory of Ordinary Differential Equations Some Applications of the Riemann Integral and the Fixed Point Theory to the Theory of Ordinary Differential and Integral Equations Mean Value Theorems for the Riemann Integral Convergence Theorems for Riemann Integrable Functions Change of Variable; Integration by Parts Improper Riemann Integrals The Lebesgue Integral Introduction Step Functions Upper Functions Lebesgue Integrable Functions Convergence Theorems Measure and Integration Functions Defined by Integrals The Space L Riemann versus Lebesgue Integrability, and the Riemann Lebesgue Criterion for Riemann Integrability The Fundamental Theorem of Calculus for Lebesgue Integration Integration by Parts Parametric Lebesgue Integrals Convex Functions Basics on Convex Functions Some Fundamental Inequalities Jensen s Inequality Using the Exponential Function Using Powers of x (Minkowski s and Hölder s Inequalities) Fourier Series Introduction Some Elementary Trigonometric Identities The Fourier Series of 2π-periodic Lebesgue Integrable Functions The Riemann Lebesgue Lemma The Partial Sums of a Fourier Series and the Dirichlet Kernel Convergence of the Fourier Series Pointwise Convergence of the Fourier Series Cesàro Convergence of the Fourier Series Uniform Convergence of the Fourier Series Convergence of the Fourier Series in Mean Square Convergence of the Fourier Series The Fourier Integral

12 Contents xv 10 Basics on Descriptive Statistics Discrete Probability Introduction Random Variables Products of Discrete Probability Spaces Inequalities Distribution Functions Selected Distributions of Discrete Random Variables Continuous Random Variables and Their Distribution Functions Excursion to Functional Analysis Real Banach Spaces Spaces with a Norm (Normed Spaces, Banach Spaces) Operators I Finite-Dimensional Banach Spaces Infinite-Dimensional Banach Spaces Operators II Finite-Rank and Compact Operators Sets of Operators Three Basic Principles of Linear Analysis Extending Continuous Linear Functionals Bounded Sets of Operators Continuity of the Inverse Operator Complex Banach Spaces The Associated Real Normed Space Operators Linear Functionals Supporting Functionals and Differentiability Basic Results in the Complex Setting Spaces with an Inner Product (Pre-Hilbertian and Hilbert Spaces) Basic Hilbert Space Theory An Application to the Uniform Convergence of the Fourier Series Complements to Hilbert Spaces Spectral Theory Pointwise Topology and Product Spaces Excursion to Nonlinear Functional Analysis Variational Principles More on Differentiability of Convex and Lipschitz Functions More on Fixed Point Theorems An Application: Periodic Distributions Introduction The Basic Idea

13 xvi Contents The Basic Definitions Derivatives of Periodic Distributions Convergence in PD Fourier Analysis Concluding Remarks to Chapter Appendix The Set of Natural Numbers Integer Numbers Rational Numbers Real Numbers The Constructive Approach The Axiomatic Approach The Complex Number System Ordering and Choice. Three Fundamental Principles in Set Theory Definitions Examples Three Basic Principles Exercises Numbers Set-Theoretical Notations Natural Numbers Fractions Base Representation Real Numbers Cardinality of Sets and Ordinal Numbers Topology of R Sequences and Series Approximation by Rational Numbers Sequences Series The Euler Number e Measure The Lebesgue Outer Measure The Class of Lebesgue Measurable Sets and the Lebesgue Measure The Cantor Ternary Set A Nonmeasurable Set Sequences of Sets Functions Functions on Real Numbers Optimization and the Mean Value Theorem The Trigonometric Functions Finer Analysis of Continuity and Differentiability

14 Contents xvii Function Convergence Function Series Metric Spaces Integration The Riemann Integral Review of Some Frequently used Techniques for calculating Antiderivatives Improper Riemann Integral Notes on Vector-Valued Riemann Integration The Lebesgue Integral Convex Functions Fourier Series Basics on Descriptive Statistics Excursion to Functional Analysis Banach Spaces Operators Finite-Dimensional Spaces Infinite-Dimensional Spaces Operators II Three Principles of Linear Analysis Spaces with an Inner Product (Pre-Hilbertian and Hilbert spaces) Spectral Theory Pointwise Topology and Product Spaces Periodic Distributions References Author Index General Index Symbol Index

15 List of Figures Fig. 1.1 Shaded, the union (i) and the intersection (ii) of families of sets 2 Fig. 1.2 A partition of a set... 2 Fig. 1.3 The hypothenuse of a right triangle, and the incommensurability with the side Fig. 1.4 The golden cut Fig. 1.5 Finding 2 by halving Fig. 1.6 The graph of the absolute value function on [ 1, 1] (Definition 37) Fig. 1.7 The positive and negative part functions Fig. 1.8 The floor and the ceiling functions Fig. 1.9 The pattern in the proof of Proposition Fig The graph {(x, f (x)) : x (0, 1)} of f (proof of Proposition 61) Fig The mapping h from ( 1, 1) onto R (proof of Proposition 61). 36 Fig The mapping g from (0, ) onto R (proof of Proposition 61). 37 Fig Catching the point x Fig The construction in the proof of Theorem 109 (sets U n in grey) 55 Fig. 2.1 How Achilles and the tortoise proceed Fig. 2.2 The partial sums B n of an alternating series and the sum B (Corollary 183) Fig. 2.3 The difference s p,q s n,m (proof of Proposition 208) Fig. 2.4 Getting a n+1,m+1 by subtracting s n+1,m s n,m from s n+1,m+1 s n,m Fig. 2.5 A particular summation method (i.e., a particular function ϕ) in Proposition Fig. 2.6 Two functions that approximate e (Proposition 216) Fig. 2.7 Inequalities (2.50) Fig. 3.1 Fig. 3.2 The first two steps in the construction of the Cantor ternary set C A tree representation of the Cantor ternary set; 0 points to the left, 1 to the right xix

16 xx List of Figures Fig. 3.3 Elements in the ε1,...,ε n intervals written using the base-3 expansion (Remark 281) Fig. 4.1 The graph of the function x 2 x + 1 on [0,1] Fig. 4.2 The graph of the function (x 1)/(x + 1) on [ 10,10] Fig. 4.3 The characteristic function of the set A Fig. 4.4 The function x 2 is even, the function x 3 is odd Fig. 4.5 The limit of a function f at a point x 0 may be different from f (x 0 ) Fig. 4.6 The function f in Example Fig. 4.7 The signum function (Eq. (4.2)) Fig. 4.8 The function x + (1/x) on[ 1, 0) (0, 1] (with partial y-range) (Example ) Fig. 4.9 At x 0, f is continuous, g discontinuous Fig The function 1/x on the interval [ 10, 0) (0, 10] (the range limited to [ 10, 10]) Fig The preimage of (1, 2) by f (x) := x 2 (Remark 324) Fig The graph of f in [ 10, 10] (proof of Corollary 332) Fig The example in Remark Fig The intermediate value theorem Fig The function x 2 on the interval [0, 1] and the argument in Example Fig The graph of x on [0, 10] (Example ) Fig The derivative of f at a is the limit of the slopes of the chords 160 Fig The closer we focus on f, the closer f looks locally as a translate of a linear function Fig The function x (Example ) Fig The function in Example 359 and its first derivative Fig A function with a local minimum and maximum at 0 (Remark 361) Fig Some local extrema of f Fig At the nonextremum point c = 0 the derivative is Fig Rolle s theorem Fig Lagrange s Mean Value Theorem Fig The function f in Remark 372, with f (0) = 1/ Fig For increasing x, slopes decrease near x 1 and increase near x Fig The plot of the Riemann function (Example 4.3) Fig The trigonometric functions Fig The trigonometric functions sin x and cos x on [ 2π,2π] Fig Adding angles α and β Fig The proof of Corollary Fig Computing the trigonometric functions at some angles Fig The function tan x and its derivative on ( π/2, π/2) Fig The function arctan x and its derivative on [-10,10] (Example 395) Fig The function arcsin x and its derivative (Example 396)

17 List of Figures xxi Fig The first steps in the construction of the Lebesgue singular function S Fig The Lebesgue singular function S (i.e., the devil s staircase) Fig One of the step functions s n (Proposition 408) Fig The function in Example 429 on [0, 2/π] Fig The function sin x as the difference of two increasing functions on [0, 2π] Fig There are few horizontal tangent lines (Lemma 440) Fig The function x and a linear function Cx (for C>0) on [0, 1] 211 Fig The graph of f and f in Example Fig The graphs of f and f (its range truncated) in Example Fig Hierarchy of some classes of functions Fig. 5.1 a The first seven functions in Example 454. b The pointwise limit of the sequence Fig. 5.2 Approximating the Riemann function Fig. 5.3 Fig. 5.4 The functions f n, after some n, are in the shaded region (uniform convergence) The first four elements in the sequence of functions in Remark Fig. 5.5 The first four functions in both sequences (Example 471) Fig. 5.6 The plot of the function f in (5.5) Fig. 5.7 A nondifferentiable uniform limit of differentiable functions Fig. 5.8 The function φ and the two first functions f 1 and f 2 (Definition 481) Fig. 5.9 Three steps in building the Takagi van der Waerden function in [0,1] Fig The graph of the Takagi van der Waerden function on [0, 1] Fig Zooming on the graph of the Takagi van der Waerden function 233 Fig The first polynomials in Lemma 484, and the limit function x 233 Fig The function exp x and its first four Taylor polynomials at Fig The functions sin x and cos x and their first six Taylor polynomials at Fig The function ln (1 + x) and its first five Taylor polynomials at Fig Examples for Corollary 507 (in all cases, x 0 = 0) Fig Building the sequence of approximations in Newton s method 249 Fig The function f and four approximations on ( 1, 1) (Example 512.1) Fig Five approximations to n=1 xn on [ 1, 1) (Example 512.2) n Fig Five approximations to n=1 xn on [ 1, 1] (Example 512.3) n 2 Fig a The first five Taylor polynomials of (1 x) 1 at x = 0. b The first four Taylor polynomials of (1 + x 2 ) 1 at x = 0 (Example ) Fig The function f in Example Fig The function in Example Fig The graph of the exponential function on the interval [ 2, 2].. 263

18 xxii List of Figures Fig The functions exp x and ln x on the interval [ 3,3] Fig Inequalities (5.81) Fig The function (1 + x)/(1 x) on (0, 1) (Remark 540) Fig Using a logarithmic table to find the product Fig Computing 2 times 3 on a slide rule Fig The hyperbolic functions sinh x, cosh x, and tanh x, on[ 5, 5] 274 Fig The trigonometric functions sin x, cos x, and tan x (its OX and Fig OY scale are different) The inverse trigonometric functions on their domains (arctan x on [ 10, 10]) Fig Graphs of (1 + x) α for several α s Fig. 6.1 Three distances in R Fig. 6.2 The distance from a point x to a set A Fig. 6.3 A uniformly continuous non-lipschitz function on [0, 1] Fig. 6.4 A homeomorphism from (0, 1) onto R (Example 562.1) Fig. 6.5 A homeomorphism from C 0 onto R (Example 562.2) Fig. 6.6 f defined on F := [0, 1], g on M := R, and K = 3 (Lemma 567) Fig. 6.7 Proof of Proposition 576: functions ϕ(x) for some x s (here M = R and x 0 = 0) Fig. 6.8 Covering a Polish space and the mapping φ (Theorem 596) Fig. 6.9 The construction in Proposition Fig The first four functions in Example Fig The first steps of the construction in Proposition 635 for finite A Fig Fig Fig Fig Approximating a function f first by a continuous piecewise linear function p and then by a function not in F n (the construction in ) The function f (x) := 1 + x from R onto R has no fixed point. The dashed line is the diagonal A continuous function from [0, 1] into itself has fixed points (Proposition 651) The graphs (in bold) of a contraction (a), and a noncontraction (b), and the iterations (6.16). The dashed line is the diagonal Fig Each f n has a fixed point, f does not Fig. 7.1 Approximating the area with inscribed rectangles Fig. 7.2 Approximating the area by using Riemann sums Fig. 7.3 Fig. 7.4 f (a)+f (b) Two functions giving the same value (but not the 2 same average!) Two approaches to the area: Upper-lower sums (solid horizontal lines) and tagged sums (dashed horizontal lines) Fig. 7.5 Upper and lower Riemann sums Fig. 7.6 The effect on the Riemann lower sum of refining the partition. 344 Fig. 7.7 The average of the function x 2 on the interval [0, 1]

19 List of Figures xxiii Fig. 7.8 Fig. 7.9 Fig Fig Fig Fig The picture for the proof of the Fundamental Theorem of Calculus for a continuous function The Fundamental Theorem of Calculus for the function f (x) = constant The Fundamental Theorem of Calculus for the function f (x) = x f R[0, 1], although F (x) := x 0 f is not differentiable The function F in Remark 686.3, and its derivative (the graph is truncated between y = 3and y = 3) The function φ on [0, 1/8] in the construction of Volterra s function Fig (iii) is the basic ingredient in building Volterra s function Fig The function F on the first central open interval (first stage) Fig First four polygonals in the Cauchy Peano construction Fig The functions ψ 0, ψ 0, and ψ 1/2 on ( 1, 1), (Remark 688) Fig The three first functions f n building the devil staircase Fig Change of variable for an increasing function G Fig The function f (x) = 1 x 2 on [0,1] Fig The function h on [ 1, 0) in Remark Fig Improper Riemann integrals of the first class Fig Improper Riemann integrals of the second class Fig Plotting sin (x)/x on [0, 50] (Example 713) Fig Fig Fig The two functions in Example 715 (part of the range of the first one, part of the range and the domain of the second one) The upper and lower Riemann sums in the proof of Theorem The function under the integral sign in Example 719, and its integral Fig The function under the integral sign in Example Fig Divisions for the Lebesgue integral Fig A step function Fig An example related to Fatou s lemma (Remark 746) Fig The three first step functions s n in Example Fig The n- and m-regularization of a function f (proof of Theorem 763) Fig A sketch of the three first functions in Remark Fig The functions F, h 2, and h 3 in the proof of Theorem Fig The graph of the function in Remark 793 and of its derivative. 427 Fig The function f (x) = x s 1 e x for s = 0.1, 0.5, 1, 2, and 3 in Example Fig The Gamma function Fig The function f (x) = ( ln (1 x))/x in Example Fig Several functions in Example Fig The function in Example Fig. 8.1 The convex hull (in grey) of a set S

20 xxiv List of Figures Fig. 8.2 The graph of a convex function on I := [ 1, 1]; the shaded region is part of its epigraph Fig. 8.3 slope(a,b) slope(a,c) slope(b,c) Fig. 8.4 A convex function discontinuous at points a and b Fig. 8.5 The argument for the boundedness for f and the existence of Fig. 8.6 limit at b (Remarks and 812.8) The unique subtangent at x 1, where f is differentiable, several subtangents at a corner x 0, where f is not Fig. 8.7 The three intervals in Remark Fig. 8.8 Two non-lipschitz convex function on [0, 1] (Remark 814) Fig. 8.9 The proof of the sufficient condition in Proposition Fig The function f (x) := x 4 is strictly convex, while f (0) = Fig The functions exp x and ln x on the interval [ 3,3] Fig Some powers x r on [0, 1] (for 0 <r<1 those functions are not convex) Fig. 9.1 Some of the functions in Lemma Fig. 9.2 Changing the variable (Remark 836) Fig. 9.3 It is (quite) clear why the Riemann Lebesgue Lemma holds Fig. 9.4 The Dirichlet kernel in [ π, π] for m = 0,1,2,3, Fig. 9.5 The graph of 1 t/2 1 sin (t/2) on [ 6, 6] (proof of Theorem 843). 468 Fig. 9.6 The 2π-periodic extension of f (x) = x + x 2 on [ π, π] Fig. 9.7 Some partial sums for the 2π-periodic expansion of f (x) = x + x 2 on [ π, π] Fig. 9.8 The 2π-periodic extension of f (x) = x + x 2 on [0, 2π] Fig. 9.9 The Fejér kernel (Definition 857)in[ π, π] for m = 0,1,2,3, Fig Two different extensions of f in Example Fig The kernel K in Remark Fig The probability density function of the random variable S Fig Two dartboards Fig The probability density and distribution functions of the two-point distribution Fig The probability density function f 20 for the binomial distribution for several p s Fig A distribution function F and its probability density function f 502 Fig The density function of a normal distribution with mean 4 and variance 3 on [ 20,20] Fig Two equivalent norms on R 2 (inclusions (11.4) Fig The first terms of a bounded sequence in X with an unbounded image by F (Example 906) Fig The closed unit ball in the norm 1 of R 3 (proof of Theorem 908) Fig The construction in Lemma

21 List of Figures xxv Fig The construction of an Auerbach basis {e i ; f i } 2 i=1 in R2 for a given norm Fig The construction of an Auerbach basis {e i ; f i } 3 i=1 in R3 for a given norm Fig The inductive construction in (1) in the proof of Theorem Fig The construction in (3) in the proof of Theorem Fig Two closed hyperplanes are always linearly isomorphic (Remark 918.3) Fig Theorem Fig The construction in the proof of Theorem Fig The construction in the proof of Corollary Fig The hyperplane H supports C at c Fig A supporting functional (Corollary 931 and Proposition 933) Fig The graph of the function (11.16)on[ 1, 1] [ 1, 1] (Example 938) Fig The graph of the function (11.17)on[ 1, 1] [ 1, 1] Fig (Example 939) Balls of a (a) non-gâteaux (b) Gâteaux differentiable norm at x 0 (Remark 940.4) Fig The norm is Fréchet differentiable at x 0 (Remark 940.5) Fig Two supporting functionals to B l 2 (at (0, 1) and (1, 1)) (Example 943) Fig In bold, the closed unit ball in l 2 4 (left) and in l2 4/3 (right), the starting point x 0 and the computed point (a, b) (Example 944). In the picture, dual balls share the same dash-style Fig The helix in Remark Fig Two projection P and Q onto Y, with P =1and Q > 1 (in gray, the image of B X ) Fig A projection of norm 1 onto the one-dimensional subspace span {x 0 } Fig The construction of a projection of norm almost 2 onto f 1 (0) 543 Fig The parallelogram equality Fig In an inner product space, the sphere does not contain segments (Remark 960) Fig In an inner produc space, a subspace F and its orthogonal complement F Fig The Pythagorean Theorem (Eq. (11.27)) Fig Searching for the point at minimum distance (Lemma 967.1) Fig Continuity of the metric projection mapping P C (1(d) in Lemma 967) Fig The closest point x 0 to x in a subspace F Fig Fig Decomposing X into an orthogonal direct sum, and the associated projections (Theorem 969 and Corollary 970) The sum of the two first summands in the Fourier series of x (Theorem 977)

22 xxvi List of Figures Fig The first elements of the Haar basis Fig The supporting functional in a real Hilbert space (Proposition 990) Fig The mappings and vectors in Corollary 994. The conclusion is that j(h ) = H Fig The norm of the Hilbert space (R 2,, 2 ) is Fréchet differentiable out of 0, and its derivative has norm Fig The function f and its perturbation (Theorem 1027) Fig Fermat s Theorem 362 almost holds (Corollary 1029) Fig The three points in the proof of Lemma Fig The construction in Remark Fig The construction of φ in Remark Fig The function in Example Fig Connections between periodic distributions and their Fourier coefficients sequence Fig The six first partial sums of the Fourier series for the distribution δ 0 (Remark 1059) (vertical scales are different) Fig The elements e ix for x R Fig Construction of the golden cut and the golden ratio (Exercise 13.24) Fig The Riemann sums in Exercise for p = 2 and n = Fig First steps in the construction of C C (Exercise ) Fig The function in Exercise Fig The function in Exercise Fig A fragment of the graph of the function 3 x (see Exercise ) Fig A fragment of the graph of the function in Exercise Fig A schema of the assumption in the hint of Exercise Fig (a) f on [0, 1] has a fixed point, (b) g on (0, 1) has no fixed points (Exercise ) Fig The function x 2 and its tangent at 1 (Exercise ) Fig The function x x on (0,1] Fig The reason why Thales circle gives the right answer (Exercise ) Fig Three functions in Exercise Fig The function in Exercise Fig The function in Exercise Fig The extension in Exercise Fig The extension in Exercise Fig A fragment of the graph of the function in Exercise Fig The functions g and h in Exercises and Fig The function ϕ in Exercise Fig The first five functions of the approximate identity in Exercise (d) Fig The function in Exercise for x 0 =

23 List of Figures xxvii Fig The function in Exercise and its asymptote at Fig The function in Exercise Fig The function in Exercise Fig The function f in Exercise Fig The function 1 3 x3 5 2 x2 + 6x + 1 on the interval [1, 4] (Exercise ) Fig The function in Exercise Fig The function f in Exercise Fig The function f and its two first derivatives on [ 1.5, 1.5] (Exercise ) Fig The first four iterates of the sinus function (Exercise ) Fig The functions tan x and x + x3 in Exercise Fig The four first functions in Exercise Fig The function x on [0, 1] (Exercises and ) Fig The graph of the three functions in Exercise Fig Extending a Lipschitz function (Exercise ) Fig Functions g 1 and g 2 in Exercise Fig Approximating a continuous function by a Lipschitz function (Exercise ) Fig Functions f 1, f 2, f 3 for C := [ 1, 1] (Exercise ) Fig The first five functions f n on [0, 3] in Exercise Fig The first six functions f n on [0, 1] in Exercise Fig The first five summands on [0, 5] in Exercise Fig The first five functions f n in Exercise Fig The first six functions f n on [0, 5] in Exercise Fig The first five functions f n and g n in Exercise Fig The function n k=1 f k in Exercise Fig The function x and its degree-3 Taylor polynomial at x 0 = (Exercise ) Fig The first four functions in the construction (Exercise ) Fig Superimposing the graphs (Exercise ) Fig Several tangent lines to e x (Exercise ) Fig Three unit balls in R Fig The distance between f and g (Exercise ) Fig The distance between Bordeaux and Marseille in the metric givenby(13.24) Fig Some elements g δ that approximate x (Exercise ) Fig The first steps in building {I s : s Z <N } (Exercise ) Fig The function T has no fixed point in (0, 1) (Exercise ) Fig A strictly metric mapping without fixed point (Exercise ) Fig The graph of ln (1 + x)/x on [ 1, 1] (Exercise ) Fig The functions f and F in Exercise Fig The function in Exercise Fig The functions f and f 2 in Exercise

24 xxviii List of Figures Fig The functions x/ x + 1 on [0, 1] (Exercise ) Fig The function under the integral sign in Exercise , for several values of a Fig The function (sinx)/x in Exercise on [0, 50] Fig The function in the Fresnel integral (Exercise ) on [0,10] Fig Comparison of functions on [0, 1] (Exercise ) Fig An improper Riemann integrable non-lebesgue integrable Fig function A step function (in bold) and its continuous approximation (dashed) Fig Several functions in Exercise Fig Some functions b n in Exercise Fig The function ln (sinx)on[0,π/2] (Exercise ) Fig Dividing the OY axis Fig The functions f and the first five ϕ n in the proof of Theorem 1083 for a particular f Fig The function in Exercise on [0, 3] sin (1/x) Fig The function x on (0, 1] (Exercise ) Fig A Lebesgue integrable function not vanishing at (Exercise ) Fig The function F in Exercise Fig The function x/(1 + x) (Exercise ) Fig A convex function is above or on any tangent (Exercise ) 793 Fig The function sin x as the difference of two convex functions (Exercise ) Fig The functions x and x on [0, 1] (Exercise ) Fig The series for f truncated at k = 4 (Exercise ) Fig A (i) strictly convex (ii) convex, continuous function on a convex compact set attains its maximum at an extreme point (Exercise ) Fig The functions f and S 4 in Exercise on [ π, π] Fig The functions f and S 3 in Exercise on [0, 2π] Fig The functions f and the first partial sums of its Fourier series (Exercise ) on [ 3π,3π] Fig The functions f and S 4 in Exercise Fig Functions t 2 at for some a s (Exercise ) Fig The inclusion of the balls B 1 := B l 2 1 and B 2 := B l 2 2 in Exercise Fig The two ellipses defining Day s norm in R 2 (Exercise ) 805 Fig Computing the distance to a hyperplane (Exercise ) Fig Decomposing T as T 3 T 2 T 1 (Exercise )

25 List of Figures xxix Fig The intersection of a finite number of hyperplanes cuts S X (Exercise ) Fig A sketch of the construction in the hint to Exercise Fig How to put any of the elements of B c0 in between two of them 817 Fig How to put a functions in B C[0,1] in between two of them Fig The extreme points do not form a closed set (Exercise ) 818 Fig (0, 0) is an extreme not exposed point of C (Exercise ). 818 Fig The construction in Exercise Fig The construction in the second part of Exercise Fig The sequence {f n } in Exercise Fig The construction in Exercise Fig A functional attains its supremum on B l 2 1 at an extreme point (Exercise )

26 List of Tables Table 13.1 Stirling formula (see Exercise and formula (13.9)) xxxi