REAL ANALYSIS (version 2.0; old here)

Suggested reading sequence

CHAPTER I: TOPOLOGY PRELIMINARIES

1. Review of basic topology concepts
2. The Concept of Convergence: Ultrafilters and Nets
3. Constructing topologies
4. Compactness
5. Locally compact spaces
6. Metric spaces
7. Baire theorem(s)

CHAPTER II: ELEMENTS OF FUNCTIONAL ANALYSIS

1. Normed vector spaces
2. Banach spaces
3. Hilbert spaces
4. The weak dual topology
5. Banach algebras
6. Banach algebras of continuous functions
7. Operator Theory on Hilbert spaces
8. Compact operators

CHAPTER III: MEASURE THEORY

1. Set arithmetic: (\sigma-)rings, (\sigma-)algebras, and monontone classes
2. Constructing (\sigma-)rings and (\sigma-)algebras
3. Measurable spaces and measurable maps
4. The concept of measure
5. Outer measures
6. The Lebesgue measure
7. Signed measures and complex measures

CHAPTER IV: INTEGRATION THEORY

1. Construction of the integral
2. Convergence theorems
3. The L^p spaces (1 \leq p < \infty)
4. Radon-Nikodym Theorems
5. The spaces L^\infty and L^{\infty,loc}
6. Duals of L^p

CHAPTER VI: PRODUCT SPACES

1. Product of two measure spaces
2. Fubini Theorem
3. Infinite products

CHAPTER VI: ANALYSIS ON LOCALLY COMPACT SPACES

1. Radon measures
2. Riesz' Theorem
3. Integration on locally compact spaces
4. Application: Borel functional calculus for normal operators

CHAPTER VII: INTRODUCTION TO HARMONIC ANALYSIS

1. Haar measure
2. The Fourier algebra
3. Group representations
4. Abelian groups: duality and Fourier transform
5. Compact groups: Peter-Weyl Theorem

APPENDICES

A. Zorn Lemma
B. Cardinal Arithmetic
C. Ordinal numbers
D. Classical Holder Inequality
E. The Hahn-Banach Theorem
F. Analytic Functions