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# Elementary Classical Analysis

Jerrold E. Marsden and Michael J. Hoffman

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
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