Al-Gwaiz / Elsanousi Elements of Real Analysis

1. Auflage 2006
ISBN: 978-1-4200-1160-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 436 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

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.

Zielgruppe

Third- and fourth-year undergraduate students of mathematics and related disciplines such as engineering, statistics, computer science, and physics.

Autoren/Hrsg.

Al-Gwaiz, M.A.

Elsanousi, S.A.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

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'Hôpital'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
INDEX