Problems and Solutions for Undergraduate Analysis

✍ Scribed by Rami Shakarchi, Serge Lang

• Publisher: Springer
• Year: 1998
• Tongue: English
• Leaves: 381
• Series: Undergraduate Texts in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Title......Page 4
Copyright Page......Page 5
Preface......Page 8
Contents......Page 10
Mappings......Page 14
Natural Numbers and Induction......Page 16
Denumerable Sets......Page 19
Equivalence Relations......Page 20
Algebraic Axioms......Page 22
Ordering Axioms......Page 23
Integers and Rational Numbers......Page 26
The Completeness Axiom......Page 28
Sequences of Numbers......Page 32
Functions and Limits......Page 35
Limits with Infinity......Page 37
Continuous Functions......Page 42
Properties of the Derivative......Page 48
Mean Value Theorem......Page 51
Inverse Functions......Page 52
Exponential......Page 56
Logarithm......Page 64
Sine and Cosine......Page 78
Complex Numbers......Page 84
Properties of the Integral......Page 86
Taylor's Formula......Page 93
Asymptotic Estimates and Stirling's Formula......Page 97
Normed Vector Spaces......Page 104
n-Space and Function Spaces......Page 109
Completenes......Page 112
Open and Closed Sets......Page 117
Basic Properties......Page 124
Continuous Maps......Page 126
Limits in Function Spaces......Page 133
Basic Properties of Compact Sets......Page 138
Continuous Maps on Compact Sets......Page 139
Relation with Open Coverings......Page 142
Series of Positive Numbers......Page 146
Non-Absolute Convergence......Page 159
Absolute and Uniform Convergence......Page 163
Power Series......Page 169
Differentiation and Integration of Series......Page 173
Approximation by Step Maps......Page 178
Properties of the Integral......Page 183
Relation Between the Integral and the Derivative......Page 192
Dirac Sequences......Page 196
The Weierstrass Theorem......Page 198
Hermitian Products and Orthogonality......Page 202
Trigonometric Polynomials as a Total Family......Page 212
Explicit·lJniform Approximation......Page 216
Pointwise Convergence......Page 221
Definition......Page 230
Criteria for Convergence......Page 232
Interchanging Derivatives and Integrals......Page 238
The Schwartz Space......Page 256
The Fourier Inversion Formula......Page 260
An Example of Fourier Transform Not in the Schwartz Space......Page 263
Partial Derivatives......Page 266
Differentiability and the Chain Rule......Page 275
Potential Functions......Page 279
Curve Integrals......Page 280
Taylor's Formula......Page 286
Maxima and the Derivative......Page 290
The Winding Number and Homology......Page 300
The Homotopy Form of the Integrability Theorem......Page 301
More on Homotopies......Page 303
The Space of Continuous Linear Maps......Page 306
The Derivative as a Linear Map......Page 308
Properties of the Derivative......Page 309
Mean Value Theorem......Page 310
The Second Derivative......Page 311
Higher Derivatives and Taylor's Formula......Page 314
The Shrinking Lemma......Page 316
Inverse Mappings, Linear Case......Page 323
The Inverse Mapping Theorem......Page 331
Product Decompositions......Page 333
Local Existence and Uniqueness......Page 340
Linear Differential Equations......Page 344
Elementary Multiple Integration......Page 350
Criteria for Admissibility......Page 356
Repeated Integrals......Page 358
Change of Variables......Page 359
Vector Fields on Spheres......Page 371
Definitions......Page 372
Inverse Image of a Form......Page 375
Stokes' Formula for Simplices......Page 376