9780387982359

This volume contains all the exercises and their solutions for Lang's second edition of UNDERGRADUATE ANALYSIS. The wide variety of exercises, which range from computational to more conceptual and which are of varying difficulty, cover the following subjects and more: real numbers, limits, continuous functions, differentiation and elementary integration, normed vector spaces, compactness, series, integration in one variable, improper integrals, convolutions, Fourier series and the Fourier integral, functions in n-space, derivatives in vector spaces, inverse and implicit mapping theorem, ordinary differential equations, multiple integrals and differential forms. This volume also serves as an independent source of problems with detailed answers beneficial for anyone interested in learning analysis. Intermediary steps and original drawings provided by the author assists students in their mastery of problem solving techniques and increases their overall comprehension of the subject matter.

Preface

vii

0 Sets and Mappings

1

(8)

0.2 Mappings

1

(2)

0.3 Natural Numbers and Induction

3

(3)

0.4 Denumerable Sets

6

(1)

0.5 Equivalence Relations

7

(2)

I Real Numbers

9

(10)

I.1 Algebraic Axioms

9

(1)

I.2 Ordering Axioms

10

(3)

I.3 Integers and Rational Numbers

13

(2)

I.4 The Completeness Axiom

15

(4)

II Limits and Continuous Functions

19

(16)

II.1 Sequences of Numbers

19

(3)

II.2 Functions and Limits

22

(2)

II.3 Limits with Infinity

24

(5)

II.4 Continuous Functions

29

(6)

III Differentiation

35

(8)

III.1 Properties of the Derivative

35

(3)

III.2 Mean Value Theorem

38

(1)

III.3 Inverse Functions

39

(4)

IV Elementary Functions

43

(30)

IV.1 Exponential

43

(8)

IV.2 Logarithm

51

(14)

IV.3 Sine and Cosine

65

(6)

IV.4 Complex Numbers

71

(2)

V The Elementary Real Integral

73

(18)

V.2 Properties of the Integral

73

(7)

V.3 Taylor's Formula

80

(4)

V.4 Asymptotic Estimates and Stirling's Formula

84

(7)

VI Normed Vector Spaces

91

(20)

VI.2 Normed Vector Spaces

91

(5)

VI.3 n-Space and Function Spaces

96

(3)

VI.4 Completeness

99

(5)

VI.5 Open and Closed Sets

104

(7)

VII Limits

111

(14)

VII.1 Basic Properties

111

(2)

VII.2 Continuous Maps

113

(7)

VII.3 Limits in Function Spaces

120

(5)

VIII Compactness

125

(8)

VIII.1 Basic Properties of Compact Sets

125

(1)

VIII.2 Continuous Maps on Compact Sets

126

(3)

VIII.4 Relation with Open Coverings

129

(4)

IX Series

133

(32)

IX.2 Series of Positive Numbers

133

(13)

IX.3 Non-Absolute Convergence

146

(4)

IX.5 Absolute and Uniform Convergence

150

(6)

IX.6 Power Series

156

(4)

IX.7 Differentiation and Integration of Series

160

(5)

X The Integral in One Variable

165

(18)

X.3 Approximation by Step Maps

165

(5)

X.4 Properties of the Integral

170

(9)

X.6 Relation Between the Integral and the Derivative

179

(4)

XI Approximation with Convolutions

183

(6)

XI.1 Dirac Sequences

183

(2)

XI.2 The Weierstrass Theorem

185

(4)

XII Fourier Series

189

(28)

XII.1 Hermitian Products and Orthogonality

189

(10)

XII.2 Trigonometric Polynomials as a Total Family

199

(4)

XII.3 Explicit Uniform Approximation

203

(5)

XII.4 Pointwise Convergence

208

(9)

XIII Improper Integrals

217

(26)

XIII.1 Definition

217

(2)

XIII.2 Criteria for Convergence

219

(6)

XIII.3 Interchanging Derivatives and Integrals

225

(18)

XIV The Fourier Integral

243

(10)

XIV.1 The Schwartz Space

243

(4)

XIV.2 The Fourier Inversion Formula

247

(3)

XIV.3 An Example of Fourier Transform Not in the Schwartz Space

250

(3)

XV Functions on n-Space

253

(34)

XV.1 Partial Derivatives

253

(9)

XV.2 Differentiability and the Chain Rule

262

(4)

XV.3 Potential Functions

266

(1)

XV.4 Curve Integrals

267

(6)

XV.5 Taylor's Formula

273

(4)

XV.6 Maxima and the Derivative

277

(10)

XVI The Winding Number and Global Potential Functions

287

(6)

XVI.2 The Winding Number and Homology

287

(1)

XVI.5 The Homotopy Form of the Integrability Theorem

288

(2)

XVI.6 More on Homotopies

290

(3)

XVII Derivatives in Vector Spaces

293

(10)

XVII.1 The Space of Continuous Linear Maps

293

(2)

XVII.2 The Derivative as a Linear Map

295

(1)

XVII.3 Properties of the Derivative

296

(1)

XVII.4 Mean Value Theorem

297

(1)

XVII.5 The Second Derivative

298

(3)

XVII.6 Higher Derivatives and Taylor's Formula

301

(2)

XVIII Inverse Mapping Theorem

303

(24)

XVIII.1 The Shrinking Lemma

303

(7)

XVIII.2 Inverse Mappings, Linear Case

310

(8)

XVIII.3 The Inverse Mapping Theorem

318

(2)

XVIII.5 Product Decompositions

320

(7)

XIX Ordinary Differential Equations

327

(10)

XIX.1 Local Existence and Uniqueness

327

(4)

XIX.3 Linear Differential Equations

331

(6)

XX Multiple Integrals

337

(22)

XX.1 Elementary Multiple Integration

337

(6)

XX.2 Criteria for Admissibility

343

(2)

XX.3 Repeated Integrals

345

(1)

XX.4 Change of Variables

346

(12)

XX.5 Vector Fields on Spheres

358

(1)

XXI Differential Forms

359

XXI.1 Definitions

359

(3)

XXI.2 Inverse Image of a Form

362

(1)

XXI.4 Stokes' Formula for Simplices

363

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Problems and Solutions for Undergraduate Analysis

0387982353

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