# Foundations of Mathematical Analysis

Courier Corporation, 2002 - Mathematics - 429 pages
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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### 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

### 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.