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Real Analysis and Foundations is an advanced undergraduate and first-year graduate textbook that introduces students to introductory topics in real analysis (or real variables), point set topology, and the calculus of variations. This classroom-tested book features over 350 end-of-chapter exercises that clearly develop and reinforce conceptual topics. It also provides an excellent review chapter on math foundations topics, as well as accessible coverage of classical topics, such as Weirstrass Approximation Theorem, Ascoli-Arzela Theorem and Schroeder-Bernstein Theorem. Explanations and discussions of key concepts are so well done that Real Analysis and Foundations will also provide valuable information for professional aerospace and structural engineers.
Author Bio
Krantz, Steven G. : Washington University
I. LOGIC AND SET THEORY.
Introduction.
"And" and "Or".
"Not" and "If-Then".
Contrapositive, Converse, and "If".
Quantifiers.
Set Theory and Venn Diagrams.
Relations and Functions.
Countable and Uncountable Sets.
II. NUMBER SYSTEMS.
The Natural Numbers.
Equivalence Relations and Equivalence Classes.
The Integers.
The Rational Numbers.
The Real Numbers.
The Complex Numbers.
III. SEQUENCES.
Convergence of Sequences.
Subsequences.
Lim sup and Lim inf.
Some Special Sequences.
IV. SERIES OF NUMBERS.
Convergence of Series.
Elementary Convergence Tests.
Advanced Convergence Tests.
Some Special Series.
Operations on Series.
V. BASIC TOPOLOGY.
Open and Closed Sets.
Further Properties of Open and Closed Sets.
Compact Sets.
The Cantor Set.
Connected and Disconnected Sets.
Perfect Sets.
VI. LIMITS AND CONTINUITY OF FUNCTIONS.
Basic Properties of the Limit of a Function.
Continuous Functions.
Topological Properties and Continuity.
Classifying Discontinuities and Monotonicity.
VII. DIFFERENTIATION OF FUNCTIONS.
The Concept of Derivative.
The Mean Value Theorem and Applications.
More on the Theory of Differentiation.
VIII. THE INTEGRAL.
Partitions and the Concept of Integral.
Properties of the Riemann Integral.
Another Look at the Integral.
Advanced Results on Integration Theory.
IX. SEQUENCES AND SERIES OF FUNCTIONS.
Partial Sums and Pointwise Convergence.
More on Uniform Convergence.
Series of Functions.
The Weirstrass Approximation Theorem.
X. SPECIAL FUNCTIONS.
Power Series.
More on Power Series.
Convergence Issues.
The Exponential and Trigonometric Functions.
Logarithms and Powers of Real Numbers.
The Gamma Function and Stirling's Formula.
An Introduction to Fourier Series.
XI. FUNCTIONS OF SEVERAL VARIABLES.
Review of Linear Algebra.
A New Look at the Basic Concepts of Analysis.
Properties of the Derivative.
The Inverse and Implicit Function Theorems.
XII. ADVANCED TOPICS.
Metric Spaces.
Topology in a Metric Space.
The Baire Category Theorem.
The Ascoli-Arzela Theorem.
Real Analysis and Foundations is an advanced undergraduate and first-year graduate textbook that introduces students to introductory topics in real analysis (or real variables), point set topology, and the calculus of variations. This classroom-tested book features over 350 end-of-chapter exercises that clearly develop and reinforce conceptual topics. It also provides an excellent review chapter on math foundations topics, as well as accessible coverage of classical topics, such as Weirstrass Approximation Theorem, Ascoli-Arzela Theorem and Schroeder-Bernstein Theorem. Explanations and discussions of key concepts are so well done that Real Analysis and Foundations will also provide valuable information for professional aerospace and structural engineers.
Author Bio
Krantz, Steven G. : Washington University
Table of Contents
I. LOGIC AND SET THEORY.
Introduction.
"And" and "Or".
"Not" and "If-Then".
Contrapositive, Converse, and "If".
Quantifiers.
Set Theory and Venn Diagrams.
Relations and Functions.
Countable and Uncountable Sets.
II. NUMBER SYSTEMS.
The Natural Numbers.
Equivalence Relations and Equivalence Classes.
The Integers.
The Rational Numbers.
The Real Numbers.
The Complex Numbers.
III. SEQUENCES.
Convergence of Sequences.
Subsequences.
Lim sup and Lim inf.
Some Special Sequences.
IV. SERIES OF NUMBERS.
Convergence of Series.
Elementary Convergence Tests.
Advanced Convergence Tests.
Some Special Series.
Operations on Series.
V. BASIC TOPOLOGY.
Open and Closed Sets.
Further Properties of Open and Closed Sets.
Compact Sets.
The Cantor Set.
Connected and Disconnected Sets.
Perfect Sets.
VI. LIMITS AND CONTINUITY OF FUNCTIONS.
Basic Properties of the Limit of a Function.
Continuous Functions.
Topological Properties and Continuity.
Classifying Discontinuities and Monotonicity.
VII. DIFFERENTIATION OF FUNCTIONS.
The Concept of Derivative.
The Mean Value Theorem and Applications.
More on the Theory of Differentiation.
VIII. THE INTEGRAL.
Partitions and the Concept of Integral.
Properties of the Riemann Integral.
Another Look at the Integral.
Advanced Results on Integration Theory.
IX. SEQUENCES AND SERIES OF FUNCTIONS.
Partial Sums and Pointwise Convergence.
More on Uniform Convergence.
Series of Functions.
The Weirstrass Approximation Theorem.
X. SPECIAL FUNCTIONS.
Power Series.
More on Power Series.
Convergence Issues.
The Exponential and Trigonometric Functions.
Logarithms and Powers of Real Numbers.
The Gamma Function and Stirling's Formula.
An Introduction to Fourier Series.
XI. FUNCTIONS OF SEVERAL VARIABLES.
Review of Linear Algebra.
A New Look at the Basic Concepts of Analysis.
Properties of the Derivative.
The Inverse and Implicit Function Theorems.
XII. ADVANCED TOPICS.
Metric Spaces.
Topology in a Metric Space.
The Baire Category Theorem.
The Ascoli-Arzela Theorem.