9780321046253

Introduction to Real Analysis

by
  • ISBN13:

    9780321046253

  • ISBN10:

    0321046250

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 11/15/2000
  • Publisher: Pearson

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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

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