Introduction to Real Analysis

by Stoll, Manfred

• ISBN13: 9780321046253
• ISBN10: 0321046250
• Edition: 2nd
• Format: Paperback
• Copyright: 2000-11-15
• Publisher: Pearson

Summary

This text is a single variable real analysis text, designed for the one-year course at the junior, senior, or beginning graduate level. It provides a rigorous and comprehensive treatment of the theoretical concepts of analysis. The book contains most of the topics covered in a text of this nature, but it also includes many topics not normally encountered in comparable texts. These include the Riemann-Stieltjes integral, the Lebesgue integral, Fourier series, the Weiestrass approximation theorem, and an introduction to normal linear spaces. The Real Number System; Sequence Of Real Numbers; Structure Of Point Sets; Limits And Continuity; Differentiation; The Riemann And Riemann-Stieltjes Integral; Series of Real Numbers; Sequences And Series Of Functions; Orthogonal Functions And Fourier Series; Lebesgue Measure And Integration; Logic and Proofs; Propositions and Connectives For all readers interested in real analysis.

Table of Contents

Preface	xi
To The Student	xv
The Real Number System	1	(46)
Sets and Operations on Sets	2	(4)
Functions	6	(9)
Mathematical Induction	15	(5)
The Least Upper Bound Property	20	(8)
Consequences of the Least upper Bound Property	28	(2)
Binary and Ternary Expansions	30	(4)
Countable and Uncountable Sets	34	(13)
Notes	43	(1)
Miscellaneous Exercises	44	(2)
Supplemental Reading	46	(1)
Sequences of Real Numbers	47	(46)
Convergent Sequences	48	(5)
Limit Theorems	53	(7)
Monotone Sequences	60	(7)
Subsequences and the Bolzano-Weierstrass Theorem	67	(6)
Limit Superior and Inferior of a Sequence	73	(7)
Cauchy Sequences	80	(6)
Series of Real Numbers	86	(7)
Notes	90	(1)
Miscellaneous Exercises	90	(2)
Supplemental Reading	92	(1)
Structure of Point Sets	93	(22)
Open and Closed Sets	93	(8)
Compact Sets	101	(6)
The Cantor Set	107	(8)
Notes	110	(1)
Miscellaneous Exercises	111	(2)
Supplemental Reading	113	(2)
Limits and Continuity	115	(50)
Limit of a Function	116	(14)
Continuous Functions	130	(14)
Uniform Continuity	144	(4)
Monotone Functions and Discontinuities	148	(17)
Notes	162	(1)
Miscellaneous Exercises	162	(1)
Supplemental Reading	163	(2)
Differentiation	165	(42)
The Derivative	166	(10)
The Mean Value Theorem	176	(14)
L'Hospital's Rule	190	(7)
Newton's Method	197	(10)
Notes	203	(1)
Miscellaneous Exercises	204	(1)
Supplemental Reading	205	(2)
The Riemann and Riemann-Stieltjes Integral	207	(72)
The Riemann Integral	208	(15)
Properties of the Riemann Integral	223	(7)
Fundamental Theorem of Calculus	230	(9)
Improper Riemann Integrals	239	(6)
The Riemann-Stieltjes Integral	245	(15)
Numerical Methods	260	(12)
Proof of Lebesgue's Theorem	272	(7)
Notes	276	(1)
Miscellaneous Exercises	277	(1)
Supplemental Reading	278	(1)
Series of Real Numbers	279	(38)
Convergence Tests	280	(14)
The Dirichlet Test	294	(5)
Absolute and Conditional Convergence	299	(7)
Square Summable Sequences	306	(11)
Notes	313	(1)
Miscellaneous Exercises	314	(1)
Supplemental Reading	315	(2)
Sequences and Series of Functions	317	(62)
Pointwise Convergence and Interchange of Limits	318	(5)
Uniform Convergence	323	(7)
Uniform Convergence and Continuity	330	(7)
Uniform Convergence and Integration	337	(2)
Uniform Convergence and Differentiation	339	(7)
The Weierstrass Approximation Theorem	346	(7)
Power Series Expansions	353	(19)
The Gamma Function	372	(7)
Notes	377	(1)
Miscellaneous Exercises	377	(1)
Supplemental Reading	378	(1)
Orthogonal Functions and Fourier Series	379	(50)
Orthogonal Functions	380	(10)
Completeness and Parseval's Equality	390	(4)
Trigonometric and Fourier Series	394	(10)
Convergence in the Mean of Fourier Series	404	(11)
Pointwise Convergence of Fourier Series	415	(14)
Notes	426	(2)
Miscellaneous Exercises	428	(1)
Supplemental Reading	428	(1)
Lebesgue Measure and Integration	429	(66)
Introduction to Measure	430	(2)
Measure of Open Sets; Compact Sets	432	(12)
Inner and Outer Measure; Measurable Sets	444	(5)
Properties of Measurable Sets	449	(6)
Measurable Functions	455	(7)
The Lebesgue Integral of a Bounded Function	462	(11)
The General Lebesgue Integral	473	(11)
Square Integrable Functions	484	(11)
Notes	491	(1)
Miscellaneous Exercises	492	(1)
Supplemental Reading	493	(2)
APPENDIX: Logic and Proofs	495	(27)
A.1 Propositions and Connectives	496	(4)
A.2 Rules of Inference	500	(7)
A.3 Mathematical Proofs	507	(8)
A.4 Use of Quantifiers	515	(7)
Supplemental Reading	521	(1)
Bibliography	522	(1)
Hints and Solutions to Selected Exercises	523	(20)
Notation Index	543	(2)
Index	545