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# Elementary Classical Analysis

## Second EditionJerrold E. Marsden, Michael J. Hoffman

©1993
Hardcover
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ISBN:9780716721055

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*Elementary Classical Analysis*strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.

## Table of Contents

**1. Introduction: Sets and Functions**

Supplement on the Axioms of Set Theory

**2. The Real Line and Euclidean Space**

Ordered Fields and the Number Systems

Completeness and the Real Number System

Least Upper Bounds

Cauchy Sequences

Cluster Points: lim inf and lim sup

Euclidean Space

Norms, Inner Products, and Metrics

The Complex Numbers

**3. Topology of Euclidean Space**

Open Sets

Interior of a Set

Closed Sets

Accumulation Points

Closure of a Set

Boundary of a Set

Sequences

Completeness

Series of Real Numbers and Vectors

**4. Compact and Connected Sets**

Compacted-ness

The Heine-Borel Theorem

Nested Set Property

Path-Connected Sets

Connected Sets

**5. Continuous Mappings**

Continuity

Images of Compact and Connected Sets

Operations on Continuous Mappings

The Boundedness of Continuous Functions of Compact Sets

The Intermediate Value Theorem

Uniform Continuity

Differentiation of Functions of One Variable

Integration of Functions of One Variable

**6. Uniform Convergence**

Pointwise and Uniform Convergence

The Weierstrass M Test

Integration and Differentiation of Series

The Elementary Functions

The Space of Continuous Functions

The Arzela-Ascoli Theorem

The Contraction Mapping Principle and Its Applications

The Stone-Weierstrass Theorem

The Dirichlet and Abel Tests

Power Series and Cesaro and Abel Summability

**7. Differentiable Mappings**

Definition of the Derivative

Matrix Representation

Continuity of Differentiable Mappings; Differentiable Paths

Conditions for Differentiability

The Chain Rule

Product Rule and Gradients

The Mean Value Theorem

Taylor's Theorem and Higher Derivatives

Maxima and Minima

**8. The Inverse and Implicit Function Theorems and Related Topics**

Inverse Function Theorem

Implicit Function Theorem

The Domain-Straightening Theorem

Further Consequences of the

Implicit Function Theorem

An Existence Theorem for Ordinary Differential Equations

The Morse Lemma

Constrained Extrema and Lagrange Multipliers

**9. Integration**

Integrable Functions

Volume and Sets of Measure Zero

Lebesgue's Theorem

Properties of the Integral

Improper Integrals

Some Convergence Theorems

Introduction to Distributions

**10. Fubini's Theorem and the Change of Variables Formula**

Introduction

Fubini's Theorem

Change of Variables Theorem

Polar Coordinates

Spherical Coordinates and Cylindrical Coordinates

A Note on the Lebesgue Integral

Interchange of Limiting Operations

**11. Fourier Analysis**

Inner Product Spaces

Orthogonal Families of Functions

Completeness and Convergence Theorems

Functions of Bounded Variation and Fejér Theory (Optional)

Computation of Fourier Series

Further Convergence Theorems

Applications

Fourier Integrals

Quantum Mechanical Formalism

**Miscellaneous Exercises**

**References**

**Answers to Selected Odd-Numbered Exercises**

**Index**