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Advanced Calculus An Introduction to Linear Analysis

ISBN-10: 0470232889
ISBN-13: 9780470232880
Edition: 2008
List price: $154.00 Buy it from $53.02
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Description: Advanced Calculus highlights the connections between calculus and linear algebra and provides a mathematically sophisticated introduction to functional analytic concepts. The book stresses that proofs must be written down, scrutinized step-by-step  More...

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Book details

List price: $154.00
Copyright year: 2008
Publisher: John Wiley & Sons, Incorporated
Publication date: 4/25/2008
Binding: Hardcover
Pages: 416
Size: 6.00" wide x 9.25" long x 1.00" tall
Weight: 1.826
Language: English

Advanced Calculus highlights the connections between calculus and linear algebra and provides a mathematically sophisticated introduction to functional analytic concepts. The book stresses that proofs must be written down, scrutinized step-by-step and rewritten whenever there is doubt. Unlike the competition, this book approaches the rigorous foundations of calculus in a manner that reorients thinking in the directions taken by modern analysis.

Leonard F. Richardson, PhD, is Herbert Huey McElveen Professor and Director of Graduate Studies in Mathematics at Louisiana State University, where he is also Assistant Chair of the Department of Mathematics. Dr. Richardson's research interests include harmonic analysis, homogeneous spaces, and representation theory. He is the author of Advanced Calculus: An Introduction to Linear Analysis, also published by Wiley.

Preface
Acknowledgments
Introduction
Advanced Calculus in One Variable
Real Numbers and Limits of Sequences
The Real Number System
Exercises
Limits of Sequences & Cauchy Sequences
Exercises
The Completeness Axiom and Some Consequences
Exercises
Algebraic Combinations of Sequences
Exercises
The Bolzano-Weierstrass Theorem
Exercises
The Nested Intervals Theorem
Exercises
The Heine-Borel Covering Theorem
Exercises
Countability of the Rational Numbers
Exercises
Test Yourself
Exercises
Continuous Functions
Limits of Functions
Exercises
Continuous Functions
Exercises
Some Properties of Continuous Functions
Exercises
Extreme Value Theorem and Its Consequences
Exercises
The Banach Space C[a, b]
Exercises
Test Yourself
Exercises
Riemann Integral
Definition and Basic Properties
Exercises
The Darboux Integrability Criterion
Exercises
Integrals of Uniform Limits
Exercises
The Cauchy-Schwarz Inequality
Exercises
Test Yourself
Exercises
The Derivative
Derivatives and Differentials
Exercises
The Mean Value Theorem
Exercises
The Fundamental Theorem of Calculus
Exercises
Uniform Convergence and the Derivative
Exercises
Cauchy's Generalized Mean Value Theorem
Exercises
Taylor's Theorem
Exercises
Test Yourself
Exercises
Infinite Series
Series of Constants
Exercises
Convergence Tests for Positive Term Series
Exercises
Absolute Convergence and Products of Series
Exercises
The Banach Space l[subscript 1] and Its Dual Space
Exercises
Series of Functions: The Weierstrass M-Test
Exercises
Power Series
Exercises
Real Analytic Functions and C[superscript infinity] Functions
Exercises
Weierstrass Approximation Theorem
Exercises
Test Yourself
Exercises
Advanced Topics in one Variable
Fourier Series
The Vibrating String and Trigonometric Series
Exercises
Euler's Formula and the Fourier Transform
Exercises
Bessel's Inequality and l[subscript 2]
Exercises
Uniform Convergence & Riemann Localization
Exercises
L[superscript 2]-Convergence & the Dual of l[superscript 2]
Exercises
Test Yourself
Exercises
The Riemann-Stieltjes Integral
Functions of Bounded Variation
Exercises
Riemann-Stieltjes Sums and Integrals
Exercises
Riemann-Stieltjes Integrability Theorems
Exercises
The Riesz Representation Theorem
Exercises
Test Yourself
Exercises
Advanced Calculus in Several Variables
Euclidean Space
Euclidean Space as a Complete Normed Vector Space
Exercises
Open Sets and Closed Sets
Exercises
Compact Sets
Exercises
Connected Sets
Exercises
Test Yourself
Exercises
Continuous Functions on Euclidean Space
Limits of Functions
Exercises
Continuous Functions
Exercises
Continuous Image of a Compact Set
Exercises
Continuous Image of a Connected Set
Exercises
Test Yourself
Exercises
The Derivative in Euclidean Space
Linear Transformations and Norms
Exercises
Differentiable Functions
Exercises
The Chain Rule in Euclidean Space
The Mean Value Theorem
Taylor's Theorem
Exercises
Inverse Functions
Exercises
Implicit Functions
Exercises
Tangent Spaces and Lagrange Multipliers
Exercises
Test Yourself
Exercises
Riemann Integration in Euclidean Space
Definition of the Integral
Exercises
Lebesgue Null Sets and Jordan Null Sets
Exercises
Lebesgue's Criterion for Riemann Integrability
Exercises
Fubini's Theorem
Exercises
Jacobian Theorem for Change of Variables
Exercises
Test Yourself
Exercises
Set Theory
Terminology and Symbols
Exercises
Paradoxes
Problem Solutions
References
Index

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