Elementary Classical Analysis

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