The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. This first volume focuses on the analysis of real-valued functions of a real variable. Besides developing the basic theory it describes many applications, including a chapter on Fourier series. It also includes a Prologue in which the author introduces the axioms of set theory and uses them to construct the real number system. Volume 2 goes on to consider metric and topological spaces and functions of several variables. Volume 3 covers complex analysis and the theory of measure and integration.

Contents for Volume II page viii

Contents for Volumne III x

Volume I

Introduction xv

Part One Prologue: The foundations of analysis 1

      1 The axioms of set theory 3

      1.1 The need for axiomatic set theory 3

      1.2 The first few axioms of set theory 5

      1.3 Relations and partial orders 9

      1.4 Functions 11

      1.5 Equivalence relations 16

      1.6 Some theorems of set theory 18

      1.7 The foundation axiom and the axiom of infinity 20

      1.8 Sequences, and recursion 23

      1.9 The axiom of choice 26

      1.10 Concluding remarks 29

      2 Number systems 32

      2.1 The non-negative integers and the natural numbers 32

      2.2 Finite and infinite sets 37

      2.3 Countable sets 42

      2.4 Sequences and subsequences 46

      2.5 The integers 49

      2.6 Divisibility and factorization 53

      2.7 The field of rational numbers 59

      2.8 Ordered fields 64

      2.9 Dedekind cuts 66

      2.10 The real number field 70

Part Two Functions of a real variable 77

      3 Convergent sequences 79

      3.1 The real numbers 79

      3.2 Convergent sequences 84

      3.3 The uniqueness of the real number system 91

      3.4 The Bolzano-Weierstrass theorem 94

      3.5 Upper and lower limits 95

      3.6 The general principle of convergence 98

      3.7 Complex numbers 99

      3.8 The convergence of complex sequences 105

      4 Infinite series 107

      4.1 Infinite series 107

      4.2 Series with non-negative terms 109

      4.3 Absolute and conditional convergence 115

      4.4 Iterated limits and iterated sums 118

      4.5 Rearranging series 120

      4.6 Convolution, or Cauchy, products 123

      4.7 Power series 126

      5 The topology of R 131

      5.1 Closed sets 131

      5.2 Open sets 135

      5.3 Connectedness 136

      5.4 Compact sets 138

      5.5 Perfect sets, and Cantor's ternary set 141

      6 Continuity 147

      6.1 Limits and convergence of functions 147

      6.2 Orders of magnitude 151

      6.3 Continuity 153

      6.4 The intermediate value theorem 162

      6.5 Point-wise convergence and uniform convergence 164

      6.6 More on power series 167

      7 Differentiation 173

      7.1 Differentiation at a point 173

      7.2 Convex functions 180

      7.3 Differentiable functions on an interval 186

      7.4 The exponential and logarithmic functions; powers 189

      7.5 The circular functions 193

      7.6 Higher derivatives, and Taylor's theorem 200

      8 Integration 209

      8.1 Elementary integrals 209

      8.2 Upper and lower Riemann integrals 211

      8.3 Riemann integrable functions 214

      8.4 Algebraic properties of the Riemann integral 220

      8.5 The fundamental theorem of calculus 223

      8.6 Some mean-value theorems 228

      8.7 Integration by parts 231

      8.8 Improper integrals and singular integrals 233

      9 Introduction to Fourier series 240

      9.1 Introduction 240

      9.2 Complex Fourier series 243

      9.3 Uniqueness 246

      9.4 Convolutions, and Parseval's equation 252

      9.5 An example 256

      9.6 The Dirichlet kernel 257

      9.7 The Fejér kernel and the Poisson kernel 264

      10 Some applications 270

      10.1 Infinite products 270

      10.2 The Taylor series of logarithmic functions 273

      10.3 The beta function 274

      10.4 Stirling's formula 277

      10.5 The gamma function 278

      10.6 Riemann's zeta function 281

      10.7 Chebyshev's prime number theorem 282

      10.8 Evaluating ζ(2) 286

      10.9 The irrationality of er 287

      10.10 The irrationality of π 289

      Appendix A Zorn's lemma and the well-ordering principle 291

      A.1 Zorn's lemma 291

      A.2 The well-ordering principle 293

Index 295

Volume II

Introduction page ix

Part Three Metric and topological spaces 301

      11 Metric spaces and normed spaces 303

      11.1 Metric spaces: examples 303

      11.2 Normed spaces 309

      11.3 Inner-product spaces 312

      11.4 Euclidean and unitary spaces 317

      11.5 Isometries 319

      11.6 *The Mazur-Ulam theorem* 323

      11.7 The orthogonal group Od 327

      12 Convergence, continuity and topology 330

      12.1 Convergence of sequences in a metric space 330

      12.2 Convergence and continuity of mappings 337

      12.3 The topology of a metric space 342

      12.4 Topological properties of metric spaces 349

      13 Topological spaces 353

      13.1 Topological spaces 353

      13.2 The product topology 361

      13.3 Product metrics 366

      13.4 Separation propertices 370

      13.5 Countability properties 375

      13.6 *Examples and counterexamples* 379

      14 Completeness 386

      14.1 Completeness 386

      14.2 Banach spaces 395

      14.3 Linear operators 400

      14.4 *Tietze's extension theorem* 406

      14.5 The completion of metric and normed spaces 408

      14.6 The contraction mapping theorem 412

      14.7 *Baire's category theoren* 420

      15 Compactness 431

      15.1 Compact topological spaces 431

      15.2 Sequentially compact topological spaces 435

      15.3 Totally bounded metric spaces 439

      15.4 Compact metric spaces 441

      15.5 Compact subsets of C(K) 445

      15.6 *The Hausdorff metric* 448

      15.7 Locally compact topological spaces 452

      15.8 Local uniform convergence 457

      15.9 Finite-dimensional normed spaces 460

      16 Connectedness 464

      16.1 Connectedness 464

      16.2 Paths and tracks 470

      16.3 Path-connectedness 473

      16.4 *Hilbert's path* 475

      16.5 *More space-filling paths* 478

      16.6 Rectifiable paths 480

Part Four Functions of a vector variable 483

      17 Differentiating functions of a vector variable 485

      17.1 Differentiating fuctions of a vector variable 485

      17.2 The mean-value inequality 491

      17.3 Partial and directional derivatives 496

      17.4 The inverse mapping theorem 500

      17.5 The implicit function theorem 502

      17.6 Higher derivatives 504

      18 Integrating functions of several variables 513

      18.1 Elementary vector-valued integrals 513

      18.2 Integrating functions of several variables 515

      18.3 Integrating vector-valued functions 517

      18.4 Repeated integration 525

      18.5 Jordan content 530

      18.6 Linear change of variables 534

      18.7 Integrating functions on Euclidean space 536

      18.8 Change of variables 537

      18.9 Differentiation under the integral sign 543

      19 Differential manifolds in Euclidean space 545

      19.1 Differential manifolds in Euclidean space 545

      19.2 Tangent vectors 548

      19.3 One-dimensional differential manifolds 552

      19.4 Lagrange multipliers 555

      19.5 Smooth partitions of unity 565

      19.6 Integration over hypersurfaces 568

      19.7 The divergence theorem 572

      19.8 Harmonic functions 582

      19.9 Curl 587

      Appendix B Linear algebra 591

      B.1 Finite-dimensional vector spaces 591

      B.2 Linear mappings and matrices 594

      B.3 Determinants 597

      B.4 Cramer's rule 599

      B.5 The trace 600

      Appendix C Exterior algebras and the cross product 601

      C.1 Exterior algebras 601

      C.2 The cross product 604

      Appendix D Tychonoff's theorem 607

Index 612

Volume III

Introduction page ix

Part Five Complex analysis 625

      20 Holomorphic functions and analytic functions 627

      20.1 Holomorphic functions 627

      20.2 The Cauchy-Riemann equations 630

      20.3 Analytic functions 635

      20.4 The exponential, logarithmic and circular functions 641

      20.5 Infinite products 645

      20.6 The maximum modulus principle 646

      21 The topology of the complex plane 650

      21.1 Winding numbers 650

      21.2 Homotopic closed paths 655

      21.3 The Jordan curve theorem 661

      21.4 Surrounding a compact connected set 667

      21.5 Simply connected sets 670

      22 Complex integration 674

      22.1 Integration along a path 674

      22.2 Approximating path integrals 680

      22.3 Cauchy's theorem 684

      22.4 The Cauchy kernel 689

      22.5 The winding number as an integral 690

      22.6 Cauchy's integral formula for circular and square paths 692

      22.7 Simply connected domains 698

      22.8 Liouville's theorem 699

      22.9 Cauchy's theorem revisited 700

      22.10 Cycles; Cauchy's integral formula revisited 702

      22.11 Functions defined inside a contour 704

      22.12 The Schwarz reflection principle 705

      23 Zeros and singularities 708

      23.1 Zeros 708

      23.2 Laurent series 710

      23.3 Isolated singularities 713

      23.4 Meromorphic functions and the complex sphere 718

      23.5 The residue theorem 720

      23.6 The principle of the argument 724

      23.7 Locating zeros 730

      24 The calculus of residues 733

      24.1 Calculating residues 733

      24.2 Integrals of the form ∫2π 0 ƒ(cost, sint) dt 734

      24.3 Integrals of the form ∫∞ -∞ ƒ(x) dx 736

      24.4 Integrals of the form ∫∞ 0 xαƒ(x) dx 742

      24.5 Integrals of the form ∫∞ 0 ƒ(x) dx 745

      25 Conformal transformations 749

      25.1 Introduction 749

      25.2 Univalent functions on C 750

      25.3 Univalent functions on the punctured plane C* 750

      25.4 The Möbius group 751

      25.5 The conformal automorphisms of D 758

      25.6 Some more conformal transformations 759

      25.7 The space H(U) of holomorphic functions on a domain U 763

      25.8 The Riemann mapping theorem 765

      26 Applications 768

      26.1 Jensen's formula 768

      26.2 The function π cot πz 770

      26.3 The functions πcosec πz 772

      26.4 Infinite products 775

      26.5 *Euler's product formula* 778

      26.6 Weierstrass products 783

      26.7 The gamma function revisited 790

      26.8 Bernoulli numbers, and the evaluation of ζ(2k) 794

      26.9 The Riemann zeta function revisited 797

Part Six Measure and Integration 801

      27 Lebesgue measure on R 803

      27.1 Introduction 803

      27.2 The size of open sets, and of closed sets 804

      27.3 Inner and outer mcasure 808

      27.4 Lebesgue measurable sets 810

      27.5 Lebesgue measure on R 812

      27.6 A non-measurable set 814

      28 Measurable spaces and measurable functions 817

      28.1 Some collections of sets 817

      28.2 Borel sets 820

      28.3 Measurable real-valued functions 821

      28.4 Measure Spaces 825

      28.5 Null sets and Borel sets 828

      28.6 Almost sure convergence 830

      29 Integration 834

      29.1 Integrating non-negative functions 834

      29.2 Integrable functions 839

      29.3 Changing measures and changing variables 846

      29.4 Convergence in measure 848

      29.5 The spaces L1 R (X, ∑, μ) and L1 C (X, ∑, μ) 854

      29.6 The spaces Lp R (X, ∑, μ) and Lp C (X, ∑, μ), for 0

      29.7 The spaces L∞ R (X, ∑, μ) and L∞ C (X, ∑, μ) 863

      30 Constructing measures 865

      30.1 Outer measures 865

      30.2 Caratheodory's extension theorem 868

      30.3 Uniqueness 871

      30.4 Product measures 873

      30.5 Borel measures on R, I 880

      31 Signed measures and complex measures 884

      31.1 Signed measures 884

      31.2 Complex measures 889

      31.3 Functions of bounded variation 891

      32 Measures on metric spaces 896

      32.1 Borel measures on metric Spaces 896

      32.2 Tight measures 898

      32.3 Radon measures 900

      33 Differentiation 903

      33.1 The Lebesgue decomposition theorem 903

      33.2 Sublinear mappings 906

      33.3 The Lebesgue differentiation theorem 908

      33.4 Borel measures on R, II 912

      34 Applications 915

      34.1 Bernstein polynomials 915

      34.2 The dual space of Lp C (X, ∑, μ )for 1≤p<∞ 918

      34.3 Convolution 919

      34.4 Fourier series revisited 924

      34.5 The Poisson kernel 927

      34.6 Boundary behaviour of harmonic fuctions 934

Index 936

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