The Way of Analysis, Revised Edition .

by Strichartz, Robert S.

• ISBN13: 9780763714970
• ISBN10: 0763714976
• Edition: Revised
• Format: Paperback
• Copyright: 2000-06-16
• Publisher: Jones & Bartlett Learning

Summary

The Way of Analysis is intended for a one- or two-semester real analysis course, including or not including an introduction to Lebesgue integration, at the undergraduate or beginning graduate level. bull; bull;Introduction to Real Analysis bull;Real Analysis I & II bull;Principles of Analysis bull;Applied Real Analysis

Table of Contents

Preface	xiii
Preliminaries	1	(24)
The Logic of Quantifiers	1	(7)
Rules of Quantifiers	1	(3)
Examples	4	(3)
Exercises	7	(1)
Infinite Sets	8	(5)
Countable Sets	8	(2)
Uncountable Sets	10	(3)
Exercises	13	(1)
Proofs	13	(5)
How to Discover Proofs	13	(4)
How to Understand Proofs	17	(1)
The Rational Number System	18	(3)
The Axiom of Choice*	21	(4)
Construction of the Real Number System	25	(48)
Cauchy Sequences	25	(13)
Motivation	25	(5)
The Definition	30	(7)
Exercises	37	(1)
The Reals as an Ordered Field	38	(12)
Defining Arithmetic	38	(3)
The Field Axioms	41	(4)
Order	45	(3)
Exercises	48	(2)
Limits and Completeness	50	(6)
Proof of Completeness	50	(2)
Square Roots	52	(2)
Exercises	54	(2)
Other Versions and Visions	56	(13)
Infinite Decimal Expansions	56	(3)
Dedekind Cuts*	59	(4)
Non-Standard Analysis*	63	(3)
Constructive Analysis*	66	(2)
Exercises	68	(1)
Summary	69	(4)
Topology of the Real Line	73	(38)
The Theory of Limits	73	(13)
Limits, Sups, and Infs	73	(5)
Limit Points	78	(6)
Exercises	84	(2)
Open Sets and Closed Sets	86	(13)
Open Sets	86	(5)
Closed Sets	91	(7)
Exercises	98	(1)
Compact Sets	99	(8)
Exercises	106	(1)
Summary	107	(4)
Continuous Functions	111	(32)
Concepts of Continuity	111	(16)
Definitions	111	(8)
Limits of Functions and Limits of Sequences	119	(2)
Inverse Images of Open Sets	121	(2)
Related Definitions	123	(2)
Exercises	125	(2)
Properties of Continuous Functions	127	(13)
Basic Properties	127	(4)
Continuous Functions on Compact Domains	131	(3)
Monotone Functions	134	(4)
Exercises	138	(2)
Summary	140	(3)
Differential Calculus	143	(58)
Concepts of the Derivative	143	(10)
Equivalent Definitions	143	(5)
Continuity and Continuous Differentiability	148	(4)
Exercises	152	(1)
Properties of the Derivative	153	(12)
Local Properties	153	(4)
Intermediate Value and Mean Value Theorems	157	(5)
Global Properties	162	(1)
Exercises	163	(2)
The Calculus of Derivatives	165	(12)
Product and Quotient Rules	165	(3)
The Chain Rule	168	(3)
Inverse Function Theorem	171	(5)
Exercises	176	(1)
Higher Derivatives and Taylor's Theorem	177	(18)
Interpretations of the Second Derivative	177	(4)
Taylor's Theorem	181	(4)
L'Hopital's Rule*	185	(3)
Lagrange Remainder Formula*	188	(2)
Orders of Zeros*	190	(2)
Exercises	192	(3)
Summary	195	(6)
Integral Calculus	201	(40)
Integrals of Continuous Functions	201	(18)
Existence of the Integral	201	(6)
Fundamental Theorems of Calculus	207	(5)
Useful Integration Formulas	212	(2)
Numerical Integration	214	(3)
Exercises	217	(2)
The Riemann Integral	219	(13)
Definition of the Integral	219	(5)
Elementary Properties of the Integral	224	(3)
Functions with a Countable Number of Discontinuities*	227	(4)
Exercises	231	(1)
Improper Integrals*	232	(4)
Definitions and Examples	232	(3)
Exercises	235	(1)
Summary	236	(5)
Sequences and Series of Functions	241	(82)
Complex Numbers	241	(9)
Basic Properties of C	241	(6)
Complex-Valued Functions	247	(2)
Exercises	249	(1)
Numerical Series and Sequences	250	(13)
Convergence and Absolute Convergence	250	(6)
Rearrangements	256	(4)
Summation by Parts*	260	(2)
Exercises	262	(1)
Uniform Convergence	263	(13)
Uniform Limits and Continuity	263	(5)
Integration and Differentiation of Limits	268	(4)
Unrestricted Convergence*	272	(2)
Exercises	274	(2)
Power Series	276	(20)
The Radius of Convergence	276	(5)
Analytic Continuation	281	(5)
Analytic Functions on Complex Domains*	286	(2)
Closure Properties of Analytic Functions*	288	(6)
Exercises	294	(2)
Approximation by Polynomials	296	(13)
Lagrange Interpolation	296	(1)
Convolutions and Approximate Identities	297	(4)
The Weierstrass Approximation Theorem	301	(4)
Approximating Derivatives	305	(2)
Exercises	307	(2)
Equicontinuity	309	(7)
The Definition of Equicontinuity	309	(3)
The Arzela-Ascoli Theorem	312	(2)
Exercises	314	(2)
Summary	316	(7)
Transcendental Functions	323	(32)
The Exponential and Logarithm	323	(14)
Five Equivalent Definitions	323	(6)
Exponential Glue and Blip Functions	329	(3)
Functions with Prescribed Taylor Expansions*	332	(3)
Exercises	335	(2)
Trigonometric Functions	337	(13)
Definition of Sine and Cosine	337	(7)
Relationship Between Sines, Cosines, and Complex Exponentials	344	(5)
Exercises	349	(1)
Summary	350	(5)
Euclidean Space and Metric Spaces	355	(64)
Structures on Euclidean Space	355	(13)
Vector Space and Metric Space	355	(3)
Norm and Inner Product	358	(6)
The Complex Case	364	(2)
Exercises	366	(2)
Topology of Metric Spaces	368	(18)
Open Sets	368	(5)
Limits and Closed Sets	373	(1)
Completeness	374	(3)
Compactness	377	(7)
Exercises	384	(2)
Continuous Functions on Metric Spaces	386	(26)
Three Equivalent Definitions	386	(5)
Continuous Functions on Compact Domains	391	(2)
Connectedness	393	(4)
The Contractive Mapping Principle	397	(2)
The Stone-Weierstrass Theorem*	399	(4)
Nowhere Differentiable Functions, and Worse*	403	(6)
Exercises	409	(3)
Summary	412	(7)
Differential Calculus in Euclidean Space	419	(40)
The Differential	419	(18)
Definition of Differentiability	419	(4)
Partial Derivatives	423	(5)
The Chain Rule	428	(4)
Differentiation of Integrals	432	(3)
Exercises	435	(2)
Higher Derivatives	437	(17)
Equality of Mixed Partials	437	(4)
Local Extrema	441	(7)
Taylor Expansions	448	(4)
Exercises	452	(2)
Summary	454	(5)
Ordinary Differential Equations	459	(56)
Existence and Uniqueness	459	(26)
Motivation	459	(8)
Picard Iteration	467	(6)
Linear Equations	473	(3)
Local Existence and Uniqueness*	476	(5)
Higher Order Equations*	481	(2)
Exercises	483	(2)
Other Methods of Solution*	485	(16)
Difference Equation Approximation	485	(5)
Peano Existence Theorem	490	(4)
Power-Series Solutions	494	(6)
Exercises	500	(1)
Vector Fields and Flows*	501	(8)
Integral Curves	501	(4)
Hamiltonian Mechanics	505	(1)
First-Order Linear P.D.E.'s	506	(1)
Exercises	507	(2)
Summary	509	(6)
Fourier Series	515	(52)
Origins of Fourier Series	515	(16)
Fourier Series Solutions of P.D.E.'s	515	(5)
Spectral Theory	520	(5)
Harmonic Analysis	525	(3)
Exercises	528	(3)
Convergence of Fourier Series	531	(31)
Uniform Convergence for C1 Functions	531	(6)
Summability of Fourier Series	537	(6)
Convergence in the Mean	543	(7)
Divergence and Gibb's Phenomenon*	550	(5)
Solution of the Heat Equation*	555	(4)
Exercises	559	(3)
Summary	562	(5)
Implicit Functions, Curves, and Surfaces	567	(56)
The Implicit Function Theorem	567	(14)
Statement of the Theorem	567	(6)
The Proof*	573	(7)
Exercises	580	(1)
Curves and Surfaces	581	(21)
Motivation and Examples	581	(4)
Immersions and Embeddings	585	(6)
Parametric Description of Surfaces	591	(6)
Implicit Description of Surfaces	597	(3)
Exercises	600	(2)
Maxima and Minima on Surfaces	602	(8)
Lagrange Multipliers	602	(3)
A Second Derivative Test*	605	(4)
Exercises	609	(1)
Arc Length	610	(8)
Rectifiable Curves	610	(4)
The Integral Formula for Arc Length	614	(2)
Arc Length Parameterization*	616	(1)
Exercises	617	(1)
Summary	618	(5)
The Lebesgue Integral	623	(68)
The Concept of Measure	623	(20)
Motivation	623	(4)
Properties of Length	627	(4)
Measurable Sets	631	(3)
Basic Properties of Measures	634	(2)
A Formula for Lebesgue Measure	636	(3)
Other Examples of Measures	639	(2)
Exercises	641	(2)
Proof of Existence of Measures*	643	(12)
Outer Measures	643	(4)
Metric Outer Measure	647	(3)
Hausdorff Measures*	650	(4)
Exercises	654	(1)
The Integral	655	(15)
Non-negative Measurable Functions	655	(5)
The Monotone Convergence Theorem	660	(4)
Integrable Functions	664	(3)
Almost Everywhere	667	(1)
Exercises	668	(2)
The Lebesgue Spaces L1 and L2	670	(12)
L1 as a Banach Space	670	(3)
L2 as a Hilbert Space	673	(3)
Fourier Series for L2 Functions	676	(5)
Exercises	681	(1)
Summary	682	(9)
Multiple Integrals	691	(36)
Interchange of Integrals	691	(14)
Integrals of Continuous Functions	691	(3)
Fubini's Theorem	694	(6)
The Monotone Class Lemma*	700	(3)
Exercises	703	(2)
Change of Variable in Multiple Integrals	705	(17)
Determinants and Volume	705	(4)
The Jacobian Factor*	709	(5)
Polar Coordinates	714	(3)
Change of Variable for Lebesgue Integrals*	717	(3)
Exercises	720	(2)
Summary	722	(5)
Index	727