Constructive Analysis

Prolog.- 1 A Constructivist Manifesto.- 1. The Descriptive Basis of Mathematics.- 2. The Idealistic Component of Mathematics.- 3. The Constructivization of Mathematics.- Notes.- 2 Calculus and the Real Numbers.- 1. Sets and Functions.- 2. The Real Number System.- 3. Sequences and Series of Real Numbers.- 4. Continuous Functions.- 5. Differentiation.- 6. Integration.- 7. Certain Important Functions.- Problems.- Notes.- 3 Set Theory.- 1. Some Basic Notions of the Theory of Sets.- 2. Complemented Sets.- 3. Neighborhood Spaces and Function Spaces.- Problems.- Notes.- 4 Metric Spaces.- 1. Fundamental Definitions and Constructions.- 2. Associated Structures.- 3. Completeness.- 4. Total Boundedness and Compactness.- 5. Spaces of Functions.- 6. Locally Compact Spaces.- Problems.- Notes.- 5 Complex Analysis.- 1. The Complex Plane.- 2. Derivatives.- 3. Integration.- 4. The Winding Number.- 5. Estimates of Size, and the Location of Zeros.- 6. Singularities and Picard's Theorem.- 7. The Riemann Mapping Theorem.- Problems.- Notes.- 6 Integration.- 1. Integration Spaces.- 2. Complete Extension of an Integral.- 3. Integrable Sets.- 4. Profiles.- 5. Positive Measures on IR.- 6. Approximation by Compact Sets.- 7. Measurable Functions.- 8. Convergence of Functions and Integrals.- 9. Product Integrals.- 10. Measure Spaces.- Problems.- Notes.- 7 Normed Linear Spaces.- 1. Definitions and Examples.- 2. Finite-Dimensional Spaces.- 3. The Lp Spaces and the Radon-Nikodym Theorem.- 4. The Extension of Linear Functionals.- 5. Quasinormed Linear Spaces; the Space L?.- 6. Dual Spaces.- 7. Extreme Points.- 8. Hilbert Space and the Spectral Theorem.- Problems.- Notes.- 8 Locally Compact Abelian Groups.- 1. Haar Measure.- 2. Convolution Operators.- 3. The Character Group.- 4. Duality and the Fourier Transform.- Problems.- Notes.- 9 Commutative Banach Algebras.- 1. Definitions and Examples.- 2. Linear Equations in a Banach Algebra.- Problems.- Notes.- References.- Symbols.