Elements of Real Analysis: Volume 284

Elements of Real Analysis: Volume 284

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Focusing on one of the main pillars of mathematics, Elements of Real Analysis provides a solid foundation in analysis, stressing the importance of two elements. The first building block comprises analytical skills and structures needed for handling the basic notions of limits and continuity in a simple concrete setting while the second component involves conducting analysis in higher dimensions and more abstract spaces. Largely self-contained, the book begins with the fundamental axioms of the real number system and gradually develops the core of real analysis. The first few chapters present the essentials needed for analysis, including the concepts of sets, relations, and functions. The following chapters cover the theory of calculus on the real line, exploring limits, convergence tests, several functions such as monotonic and continuous, power series, and theorems like mean value, Taylor's, and Darboux's. The final chapters focus on more advanced theory, in particular, the Lebesgue theory of measure and integration. Requiring only basic knowledge of elementary calculus, this textbook presents the necessary material for a first course in real analysis. Developed by experts who teach such courses, it is ideal for undergraduate students in mathematics and related disciplines, such as engineering, statistics, computer science, and physics, to understand the foundations of real analysis.show more

Product details

  • Hardback | 436 pages
  • 162.6 x 236.2 x 30.5mm | 748.44g
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 60 black & white illustrations
  • 1584886617
  • 9781584886617

Table of contents

PREFACE PRELIMINARIES Sets Functions REAL NUMBERS Field Axioms Order Axioms Natural Numbers, Integers, Rational Numbers Completeness Axiom Decimal Representation of Real Numbers Countable Sets SEQUENCES Sequences and Convergence Properties of Convergent Sequences Monotonic Sequences The Cauchy Criterion Subsequences Upper and Lower Limits Open and Closed Sets INFINITE SERIES Basic Properties Convergence Tests LIMIT OF A FUNCTION Limit of a Function Basic Theorems Some Extensions of the Limit Monotonic Functions CONTINUITY Continuous Functions Combinations of Continuous Functions Continuity on an Interval UniformContinuity Compact Sets and Continuity DIFFERENTIATION The Derivative TheMean Value Theorem L'Hopital's Rule Taylor's Theorem THE RIEMANN INTEGRAL Riemann Integrability Darboux's Theorem and Riemann Sums Properties of the Integral The Fundamental Theorem of Calculus Improper Integrals SEQUENCES AND SERIES OF FUNCTIONS Sequences of Functions Properties of Uniform Convergence Series of Functions Power Series LEBESGUE MEASURE Classes of Subsets of R Lebesgue Outer Measure Lebesgue Measure Measurable Functions LEBESGUE INTEGRATION Definition of the Lebesgue Integral Properties of the Lebesgue Integral Lebesgue Integral and Pointwise Convergence Lebesgue and Riemann Integrals REFERENCES NOTATION INDEXshow more

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