Foundations of Mathematical Analysis

by Johnsonbaugh, Richard; Pfaffenberger, W.E.

• ISBN13: 9780486421742
• ISBN10: 0486421740
• Edition: Reprint
• Format: Paperback
• Copyright: 2002-08-06
• Publisher: Dover Publications

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