Foundations of Mathematical Analysis

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Courier Dover Publications, 2002 - Mathematics - 429 pages
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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 edition. 34 figures.
  

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User Review  - Stephen - Goodreads

This has been such a critical book in my mathematical development. I find myself often referring back to this book as a source to fill in holes, and referring it to colleagues as a good book to do so Read full review

Contents

Sets and Functions
1
The Real Number System
9
Set Equivalence
26
Sequences of Real Numbers
34
Real Exponents
55
The BolzanoWei erst rass Theorem
58
The Cauchy Condition
59
The lim sup and lirn inf of Bounded Sequences
61
Closed Sets
128
Open Sets
132
Continuous Functions on Metric Spaces
136
The Relative Metric
141
Compact Metric Spaces
144
The BoizanoWeierstrass Characterization of a Compact Metric Space
148
Continuous Functions on Compact Metric Spaces
152
Connected Metric Spaces
155

The lim sup and lim inf of Unbounded Sequences
69
Infinite Series
73
Algebraic Operations on Series
76
Series with Nonnegative Terms
77
The Alternating Series Test
80
Absolute Convergence
81
Power Series
87
Conditional Convergence
90
Double Series and Applications
92
Limits of RealValued Functions and Continuous Functions on the Real Line
102
Limit Theorems for Functions
105
OneSided and Infinite Limits
107
Continuity
109
The HeineBorel Theorem and a Consequence for Continuous Functions
112
VH Metric Spaces
116
R I2 and the CauchySchwarz Inequality
120
Sequences in Metric Spaces
125
Complete Metric Spaces
159
Baire Category Theorem
166
Differential Calculus of the Real Line
171
The RiemannStieltjes Integral
189
Sequences and Series of Functions
245
Transcendental Functions
268
Inner Product Spaces and Fourier Series
280
Normed Linear Spaces and the Riesz Representation
335
The Dual Space of a Normed Linear Space
343
Proof of the Riesz Representation Theorem
349
The Lebesgue Integral
355
Vector Spaces
405
Hints to Selected Exercises
411
Index
421
Errata
429
Copyright

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About the author (2002)

Richard Johnsonbaugh" has a Ph.D. from the University of Oregon. He is professor of Computer Science and Information Systems, at DePaul University. He has 25 years of experience in teaching and research, including programming in general and in the C language. Dr. Johnsonbaugh specializes in programming languages, compilers, data structures, and pattern recognition. He is the author of two very successful books on Discrete Mathematics.

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