Foundations of Mathematical Analysis,9780486421742

Foundations of Mathematical Analysis

by ;
Edition: Reprint
ISBN13: 9780486421742
ISBN10: 0486421740
Format: Paperback
Pub. Date: 2002-08-06
Publisher(s): Dover Publications
List Price: $24.56

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Summary

This classroom-tested volume offers a definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. A self-contained text, it presents the necessary background on the limit concept. (The first seven chapters could constitute a one-semester course on introduction to limits.) Subsequent chapters discuss differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue integral. More than 750 exercises help reinforce the material. 1981 ed. 34 Figures.

Table of Contents

Preface iii
Preface to the Dover Edition v
I Sets and Functions 1(8)
Sets
1(3)
Functions
4(5)
II The Real Number System 9(17)
The Algebraic Axioms of the Real Numbers
9(3)
The Order Axiom of the Real Numbers
12(2)
The Least-Upper-Bound Axiom
14(3)
The Set of Positive Integers
17(3)
Integers, Rationals, and Exponents
20(6)
III Set Equivalence 26(8)
Definitions and Examples
26(3)
Countable and Uncountable Sets
29(5)
IV Sequences of Real Numbers 34(39)
Limit of a Sequence
34(4)
Subsequences
38(2)
The Algebra of Limits
40(5)
Bounded Sequences
45(1)
Further Limit Theorems
46(2)
Divergent Sequences
48(1)
Monotone Sequences and the Number e
49(6)
Real Exponents
55(3)
The Bolzano-Weierstrass Theorem
58(1)
The Cauchy Condition
59(2)
The lim sup and lim inf of Bounded Sequences
61(8)
The lim sup and lim inf of Unbounded Sequences
69(4)
V Infinite Series 73(29)
The Sum of an Infinite Series
73(3)
Algebraic Operations on Series
76(1)
Series with Nonnegative Terms
77(3)
The Alternating Series Test
80(1)
Absolute Convergence
81(6)
Power Series
87(3)
Conditional Convergence
90(2)
Double Series and Applications
92(10)
VI Limits of Real-Valued Functions and Continuous Functions on the Real Line 102(14)
Definition of the Limit of a Function
102(3)
Limit Theorems for Functions
105(2)
One -Sided and Infinite Limits
107(2)
Continuity
109(3)
The Heine-Borel Theorem and a Consequence for Continuous Functions
112(4)
VII Metric Spaces 116(55)
The Distance Function
116(4)
Rn, l2, and the Cauchy-Schwarz Inequality
120(5)
Sequences in Metric Spaces
125(3)
Closed Sets
128(4)
Open Sets
132(4)
Continuous Functions on Metric Spaces
136(5)
The Relative Metric
141(3)
Compact Metric Spaces
144(4)
The Bolzano-Weierstrass Characterization of a Compact Metric Space
148(4)
Continuous Functions on Compact Metric Spaces
152(3)
Connected Metric Spaces
155(4)
Complete Metric Spaces
159(7)
Baire Category Theorem
166(5)
VIII Differential Calculus of the Real Line 171(18)
Basic Definitions and Theorems
171(5)
Mean-Value Theorems and L'Hospital's Rule
176(9)
Taylor's Theorem
185(4)
IX The Riemann-Stieltjes Integral 189(56)
Riemann-Stieltjes Integration with Respect to an Increasing Integrator
190(14)
Riemann-Stieltjes Sums
204(6)
Riemann-Stieltjes Integration with Respect to an Arbitrary Integrator
210(3)
Functions of Bounded Variation
213(6)
Riemann-Stieltjes Integration with Respect to Functions of Bounded Variation
219(6)
The Riemann Integral
225(5)
Measure Zero
230(4)
A Necessary and Sufficient Condition for the Existence of the Riemann Integral
234(4)
Improper Riemann-Stieltjes Integrals
238(7)
X Sequences and Series of Functions 245(23)
Pointwise Convergence and Uniform Convergence
245(4)
Integration and Differentiation of Uniformly Convergent Sequences
249(4)
Series of Functions
253(6)
Applications to Power Series
259(3)
Abel's Limit Theorems
262(3)
Summability Methods and Tauberian Theorems
265(3)
XI Transcendental Functions 268(12)
The Exponential Function
268(3)
The Natural Logarithm Function
271(3)
The Trigonometric Functions
274(6)
XII Inner Product Spaces and Fourier Series 280(55)
Normed Linear Spaces
280(5)
The Inner Product Space R3
285(3)
Inner Product Spaces
288(5)
Orthogonal Sets in Inner Product Spaces
293(2)
Periodic Functions
295(3)
Fourier Series: Definition and Examples
298(4)
Orthonormal Expansions in Inner Product Spaces
302(6)
Pointwise Convergence of Fourier Series in R[a, a + 2π]
308(7)
Cesaro Summability of Fourier Series
315(7)
Fourier Series in R[a, a + 2π]
322(9)
A Tauberian Theorem and an Application to Fourier Series
331(4)
XIII Normed Linear Spaces and the Riesz Representation Theorem 335(20)
Normed Linear Spaces and Continuous Linear Transformations
335(4)
The Normed Linear Space of Continuous Linear Transformations
339(4)
The Dual Space of a Normed Linear Space
343(3)
Introduction to the Riesz Representation Theorem
346(3)
Proof of the Riesz Representation Theorem
349(6)
XIV The Lebesgue Integral 355(50)
The Extended Real Line
356(1)
σ-Algebras and Positive Measures
357(4)
Measurable Functions
361(7)
Integration on Positive Measure Spaces
368(13)
Lebesgue Measure on R
381(11)
Lebesgue Measure on [a, b]
392(5)
The Hilbert Spaces L2(X,M,μ)
397(8)
Appendix: Vector Spaces 405(4)
References 409(2)
Hints to Selected Exercises 411(10)
Index 421(8)
Errata 429

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