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Constructive Real Analysis

Table of Contents

• Preface
• CHAPTER I/ ROOTS AND EXTREMAL PROBLEMS
  • INTRODUCTION
  • SECTION A. Iterations and Fixed Points
    • A-1. Functional Iteration and Roots: One Variable
    • A-2. Newton's Method
    • A-3. Subcontractors
  • SECTION B. Metric Spaces
    • B-1. Definitions
    • B-2. Review
    • B-3. More Definitions and Information from Analysis
    • B-4. Contraction Mapping Theorem
    • B-5. Subcontraction Mapping Theorem
  • SECTION C. Miscellany
    • C-1. Definitions
    • C-2. Norms
    • C-3. Generalized Mean-Value Theorem
    • C-4. Spectral Bounds
    • C-5. Minimization of Some Functions
  • SECTION D. Gradient Techniques
    • D-1. Heuristic Remarks
    • D-2. Gradient Method
    • D-3. Steepest Descent
    • D-4. Acceleration
    • D-5. Semicontinuity
    • D-6. Roots of Systems of Equations
    • D-7. Application to Linear Approximation
• CHAPTER II / CONSTRAINTS
  • SECTION A. Nonlinear Programming
    • A-1. Constraints and Penalty Functions
    • A-2. Extrema on Spheres and Supporting Hyperplanes for Convex Sets
  • SECTION B. Polyhedral Convex Programming
    • B-1. On Homogeneous Linear Inequalities
    • B-2. Polyhedral Convex Programming
    • B-3. Implementation of the Algorithm
  • SECTION C. Infinite Convex Programming
    • C-1. Nonpolyhedral Convex Programming I
    • C-2. Nonpolyhedral Convex Programming II
• CHAPTER III / INFINITE DIMENSIONAL PROBLEMS
  • SECTION A. Linear Spaces and Convex Sets
    • A-1. Linear Spaces
    • A-2. Normed Linear Spaces
    • A-3. Hilbert Space
    • A-4. Convex Sets in Hilbert Space
    • A-5. Projection Operator for Convex Sets
    • A-6. Distance Between Polytopes
    • A-7. On Linear Inequalities
  • SECTION B. Miscellany
    • B-1. Linear Operators
    • B-2. Application to Mechanical Quadrature
    • B-3. The Conjugate of a Hilbert Space
    • B-4. The Frechet Differential
    • B-5. The Gateaux Differential
    • B-6. The "Chain Rule"
    • B-7. Taylor's Formula for Twice G-Differentiable Real-Valued Functions
    • B-8. Weak Convergence
    • B-9. Weak Compactness Theorem
    • B-10. Characterization of Extremals in Convex Programming
    • B-11. Convex Programming
    • B-12. Rate of Convergence
  • SECTION C. Roots and Extremals
    • C-1. Theorem 1 (Hahn-Banach)
    • C-2. Mean-Value Theorem
    • C-3. Reflexive Spaces, Locally Uniformly Convex Spaces, and Inverse Operators
    • C-4. Newton's Method
    • C-5. Minimizing Functionals on NLS
  • SECTION D. Applications to Integral Equations
    • D-1. Resolvent Kernel
    • D-2. Solution by Gradient Method
    • D-3. Nonlinear Integral Equations
  • SECTION E. An Application to Control Theory
    • E-1. Rendezvous Problem
    • E-2. Application of Convex Programming
• Notes and Bibliographic Material
• Index

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