Analysis: An Introduction

by Richard Beals

• ISBN13: 9780521600477
• ISBN10: 0521600472
• Format: Paperback
• Copyright: 2004-09-13
• Publisher: Cambridge University Press

Summary

This self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics.

Author Biography

Richard Beals is James E. English Professor of Mathematics at Yale University.

Table of Contents

Preface	ix
1 Introduction	1	(14)
1A. Notation and Motivation	1	(4)
1B. The Algebra of Various Number Systems	5	(4)
1C. The Line and Cuts	9	(3)
1D. Proofs, Generalizations, Abstractions, and Purposes	12	(3)
2 The Real and Complex Numbers	15	(15)
2A. The Real Numbers	15	(6)
2B. Decimal and Other Expansions; Countability	21	(3)
2C. Algebraic and Transcendental Numbers	24	(2)
2D. The Complex Numbers	26	(4)
3 Real and Complex Sequences	30	(15)
3A. Boundedness and Convergence	30	(3)
3B. Upper and Lower Limits	33	(2)
3C. The Cauchy Criterion	35	(2)
3D. Algebraic Properties of Limits	37	(2)
3E. Subsequences	39	(1)
3F. The Extended Reals and Convergence to ± infinity	40	(2)
3G. Sizes of Things: The Logarithm	42	(1)
Additional Exercises for Chapter 3	43	(2)
4 Series	45	(16)
4A. Convergence and Absolute Convergence	45	(3)
4B. Tests for (Absolute) Convergence	48	(6)
4C. Conditional Convergence	54	(3)
4D. Euler's Constant and Summation	57	(1)
4E. Conditional Convergence: Summation by Parts	58	(1)
Additional Exercises for Chapter 4	59	(2)
5 Power Series	61	(12)
5A. Power Series, Radius of Convergence	61	(2)
5B. Differentiation of Power Series	63	(3)
5C. Products and the Exponential Function	66	(4)
5D. Abel's Theorem and Summation	70	(3)
6 Metric Spaces	73	(13)
6A. Metrics	73	(2)
6B. Interior Points, Limit Points, Open and Closed Sets	75	(4)
6C. Coverings and Compactness	79	(2)
6D. Sequences, Completeness, Sequential Compactness	81	(3)
6E. The Cantor Set	84	(2)
7 Continuous Functions	86	(13)
7A. Definitions and General Properties	86	(4)
7B. Real- and Complex-Valued Functions	90	(1)
7C. The Space C(I)	91	(4)
7D. Proof of the Weierstrass Polynomial Approximation Theorem	95	(4)
8 Calculus	99	(20)
8A. Differential Calculus	99	(6)
8B. Inverse Functions	105	(2)
8C. Integral Calculus	107	(5)
8D. Riemann Sums	112	(1)
8E. Two Versions of Taylor's Theorem	113	(3)
Additional Exercises for Chapter 8	116	(3)
9 Some Special Functions	119	(12)
9A. The Complex Exponential Function and Related Functions	119	(5)
9B. The Fundamental Theorem of Algebra	124	(1)
9C. Infinite Products and Euler's Formula for Sine	125	(6)
10 Lebesgue Measure on the Line	131	(13)
10A. Introduction	131	(2)
10B. Outer Measure	133	(3)
1OC. Measurable Sets	136	(3)
10D. Fundamental Properties of Measurable Sets	139	(3)
10E. A Nonmeasurable Set	142	(2)
11 Lebesgue Integration on the Line	144	(14)
11A. Measurable Functions	144	(4)
11B. Two Examples	148	(1)
11C. Integration: Simple Functions	149	(2)
11D. Integration: Measurable Functions	151	(4)
11E. Convergence Theorems	155	(3)
12 Function Spaces	158	(15)
12A. Null Sets and the Notion of "Almost Everywhere"	158	(1)
12B. Riemann Integration and Lebesgue Integration	159	(3)
12C. The Space L¹	162	(4)
12D. The Space L²	166	(2)
12E. Differentiating the Integral	168	(4)
Additional Exercises for Chapter 12	172	(1)
13 Fourier Series	173	(24)
13A. Periodic Functions and Fourier Expansions	173	(3)
13B. Fourier Coefficients of Integrable and Square-Integrable Periodic Functions	176	(4)
13C. Dirichlet's Theorem	180	(4)
13D. Fejér's Theorem	184	(3)
13E. The Weierstrass Approximation Theorem	187	(2)
13F. L²-Periodic Functions: The Riesz-Fischer Theorem	189	(3)
13G. More Convergence	192	(3)
13H. Convolution	195	(2)
14 Applications of Fourier Series	197	(21)
14A. The Gibbs Phenomenon	197	(2)
14B. A Continuous, Nowhere Differentiable Function	199	(1)
14C. The Isoperimetric Inequality	200	(2)
14D. Weyl's Equidistribution Theorem	202	(1)
14E. Strings	203	(4)
14F. Woodwinds	207	(2)
14G. Signals and the Fast Fourier Transform	209	(2)
14H. The Fourier Integral	211	(4)
14I. Position, Momentum, and the Uncertainty Principle	215	(3)
15 Ordinary Differential Equations	218	(19)
15A. Introduction	218	(1)
15B. Homogeneous Linear Equations	219	(4)
15C. Constant Coefficient First-Order Systems	223	(4)
15D. Nonuniqueness and Existence	227	(3)
15E. Existence and Uniqueness	230	(4)
15F. Linear Equations and Systems, Revisited	234	(3)
Appendix: The Banach-Tarski Paradox	237	(4)
Hints for Some Exercises	241	(14)
Notation Index	255	(2)
General Index	257