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Description:

Bridging the gap between calculus and further abstract topics this book presents a well organized and much needed introduction to the foundations of analysis. It is composed of three sections: the analysis of functions of one real variable, including an introduction to the Lebesgue integral; how the appropriate abstractions lead to a powerful and widely applicable theoretical foundation for all branches of applied mathematics; an outlook to applied subjects in which analysis is used.

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Book details

List price: $128.00 Copyright year: 2008 Publisher: John Wiley & Sons, Incorporated Publication date: 11/12/2007 Binding: Hardcover Pages: 584 Size: 6.55" wide x 9.55" long x 1.25" tall Weight: 2.068 Language: English

AuthorTable of Contents

Table of Contents

Preface

Analysis of Functions of a Single Real Variable

The Real Numbers

Field Axioms

Order Axioms

Lowest Upper and Greatest Lower Bounds

Natural Numbers, Integers, and Rational Numbers

Recursion, Induction, Summations, and Products

Sequences of Real Numbers

Limits

Limit Laws

Cauchy Sequences

Bounded Sequences

Infinite Limits

Continuous Functions

Limits of Functions

Limit Laws

One-Sided Limits and Infinite Limits

Continuity

Properties of Continuous Functions

Limits at Infinity

Differentiable Functions

Differentiability

Differentiation Rules

Rolle's Theorem and the Mean Value Theorem

The Riemann Integral I

Riemann Sums and the Integral

Uniform Continuity and Integrability of Continuous Functions

The Fundamental Theorem of Calculus

The Darboux Integral

Series of Real Numbers I

Series as a Vehicle To Define Infinite Sums

Absolute Convergence and Unconditional Convergence

Some Set Theory

The Algebra of Sets

Countable Sets

Uncountable Sets

The Riemann Integral II

Outer Lebesgue Measure

Lebesgue's Criterion for Riemann Integrability

More Integral Theorems

Improper Riemann Integrals

The Lebesgue Integral

Lebesgue Measurable Sets

Lebesgue Measurable Functions

Lebesgue Integration

Lebesgue Integrals versus Riemann Integrals

Series of Real Numbers II

Limits Superior and Inferior

The Root Test and the Ratio Test

Power Series

Sequences of Functions

Notions of Convergence

Uniform Convergence

Transcendental Functions

The Exponential Function

Sine and Cosine

L'Hopital's Rule

Numerical Methods

Approximation with Taylor Polynomials

Newton's Method

Numerical Integration

Analysis in Abstract Spaces

Integration on Measure Spaces

Measure Spaces

Outer Measures

Measurable Functions

Integration of Measurable Functions

Monotone and Dominated Convergence

Convergence in Mean, in Measure, and Almost Everywhere

Product [sigma]-Algebras

Product Measures and Fubini's Theorem

The Abstract Venues for Analysis

Abstraction I: Vector Spaces

Representation of Elements: Bases and Dimension

Identification of Spaces: Isomorphism

Abstraction II: Inner Product Spaces

Nicer Representations: Orthonormal Sets

Abstraction III: Normed Spaces

Abstraction IV: Metric Spaces

L[superscript p] Spaces

Another Number Field: Complex Numbers

The Topology of Metric Spaces

Convergence of Sequences

Completeness

Continuous Functions

Open and Closed Sets

Compactness

The Normed Topology of R[superscript d]

Dense Subspaces

Connectedness

Locally Compact Spaces

Differentiation in Normed Spaces

Continuous Linear Functions

Matrix Representation of Linear Functions

Differentiability

The Mean Value Theorem

How Partial Derivatives Fit In

Multilinear Functions (Tensors)

Higher Derivatives

The Implicit Function Theorem

Measure, Topology, and Differentiation

Lebesgue Measurable Sets in R[superscript d]

C[infinity] and Approximation of Integrable Functions

Tensor Algebra and Determinants

Multidimensional Substitution

Introduction to Differential Geometry

Manifolds

Tangent Spaces and Differentiable Functions

Differential Forms, Integrals Over the Unit Cube

k-Forms and Integrals Over k-Chains

Integration on Manifolds

Stokes' Theorem

Hilbert Spaces

Orthonormal Bases

Fourier Series

The Riesz Representation Theorem

Applied Analysis

Physics Background

Harmonic Oscillators

Heat and Diffusion

Separation of Variables, Fourier Series, and Ordinary Differential Equations

Maxwell's Equations

The Navier Stokes Equation for the Conservation of Mass

Ordinary Differential Equations

Banach Space Valued Differential Equations

An Existence and Uniqueness Theorem

Linear Differential Equations

The Finite Element Method

Ritz-Galerkin Approximation

Weakly Differentiable Functions

Sobolev Spaces

Elliptic Differential Operators

Finite Elements

Conclusion and Outlook

Appendices

Logic

Statements

Negations

Set Theory

The Zermelo-Fraenkel Axioms

Relations and Functions

Natural Numbers, Integers, and Rational Numbers

The Natural Numbers

The Integers

The Rational Numbers

Bibliography

Index

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