"Real Analysis 1 Class Notes" Webpage

Real Analysis Class Notes
Real Analysis, 4th Edition, H. L. Royden and P.M. Fitzpatrick. Royden's Real Analysis book, 4th Edition

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses.

I. Lebesgue Integration for Functions of a Single Variable.

Background and Motivation.

Supplement. Introduction to Math Philosophy and Meaning. Introduction to Math Philosophy and Meaning notes
Supplement. Essential Background for Real Analysis I. Essential Background for Real Analysis I notes
Supplement. The Riemann-Lebesgue Theorem. The Riemann-Lebesgue Theorem notes

1. The Real Numbers: Sets, Sequences, and Functions.

Section 1.4. Open Sets, Closed Sets, and Borel Sets of Real Numbers. Section 1.4 notes

Supplement. Proofs of Theorems in Section 1.4. Beamer file of Section 1.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 1.4. Print file of Section 1.4 proofs

Study Guide 1

2. Lebesgue Measure.

2.1. Introduction. Section 2.1 notes

Supplement. Proofs of Theorems in Section 2.1. Beamer file of Section 2.1 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.1. Print file of Section 2.1 proofs

2.2. Lebesgue Outer Measure. Section 2.2 notes

Supplement. Proofs of Theorems in Section 2.2. Beamer file of Section 2.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.2. Print file of Section 2.2 proofs

2.3. The σ-Algebra of Lebesgue Measurable Sets. Section 2.3 notes

Supplement. Proofs of Theorems in Section 2.3. Beamer file of Section 2.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.3. Print file of Section 2.3 proofs

Supplement. An Alternate Approach to the Measure of a Set of Real Numbers Alternate Approach to Measure notes.
2.4. Outer and Inner Approximation of Lebesgue Measurable Sets. Section 2.4 notes

Supplement. Proofs of Theorems in Section 2.4. Beamer file of Section 2.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.4. Print file of Section 2.4 proofs

2.5. Countable Additivity, Continuity, and the Borel-Cantelli Lemma. Section 2.5 notes

Supplement. Proofs of Theorems in Section 2.5. Beamer file of Section 2.5 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.5. Print file of Section 2.5 proofs

Supplement. Dr. Bob's Axiom of Choice Centennial Lecture. Axiom of Choice Centennial notes
Supplement. 2.6. Nonmeasurable Sets (from Royden's 3rd Edition). Section 2.6 notes from the 3rd edition

Supplement. Proofs of Theorems in Section 2.6 (3rd Ed.). Beamer file of Section 2.6 (3rd Ed.) proofs
Supplement. Printout of the Proofs of Theorems in Section 2.6 (3rd Ed.). Print file of Section 2.6 (3rd Ed.) proofs

2.6. Nonmeasurable Sets (from Royden and Fitzpatrick's 4th Edition). Section 2.6 notes

Supplement. Proofs of Theorems in Section 2.6 (4th Ed.). Beamer file of Section 2.6 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.6 (4th Ed.). Print file of Section 2.6 proofs

Supplement. Nonmeasurable Sets and the Banach-Tarski Paradox. Banach-Tarski Paradox notes
2.7. The Cantor Set and the Cantor Lebesgue Function. Section 2.7 notes

Supplement. Proofs of Theorems in Section 2.7. Beamer file of Section 2.7 proofs
Supplement. Printout of the Proofs of Theorems in Section 2.7. Print file of Section 2.7 proofs

Supplement. The Cardinalities of the Set of Measurable Sets and the Set of Nonmeasurable Sets. Cardinalities of Sets of Measurable and Nonmeasurable Sets notes
Study Guide 2

3. Lebesgue Measurable Functions.

3.1. Sums, Products, and Compositions. Section 3.1 notes

Supplement. Proofs of Theorems in Section 3.1. Beamer file of Section 3.1 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.1. Print file of Section 3.1 proofs

3.2. Sequential Pointwise Limits and Simple Approximation. Section 3.2 notes

Supplement. Proofs of Theorems in Section 3.2. Beamer file of Section 3.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.2. Print file of Section 3.2 proofs

3.3. Littlewood's Three Principals, Egoroff's Theorem, and Lusin's Theorem. Section 3.3 notes

Supplement. Proofs of Theorems in Section 3.3. Beamer file of Section 3.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.3. Print file of Section 3.2 proofs

Study Guide 3

4. Lebesgue Integration.

4.1. The Riemann Integral. Section 4.1 notes
4.2. The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure. Section 4.2 notes

Supplement. Proofs of Theorems in Section 4.2. Beamer file of Section 4.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.2. Print file of Section 4.2 proofs

4.3. The Lebesgue Integral of a Measurable Nonnegative Function. Section 4.3 notes

Supplement. Proofs of Theorems in Section 4.3. Beamer file of Section 4.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.3. Print file of Section 4.3 proofs

Supplement. The Dirac Delta Function - A Cautionary Tale. Dirac Delta Function notes
4.4. The General Lebesgue Integral. Section 4.4 notes

Supplement. Proofs of Theorems in Section 4.4. Beamer file of Section 4.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.4. Print file of Section 4.4 proofs

4.5. Countable Additivity and Continuity of Integration. Section 4.5 notes

Supplement. Proofs of Theorems in Section 4.5. Beamer file of Section 4.5 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.5. Print file of Section 4.5 proofs

4.6. Uniform Integrability: The Vitali Convergence Theorem. Section 4.6 notes

Supplement. Proofs of Theorems in Section 4.6. Beamer file of Section 4.6 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.6. Print file of Section 4.6 proofs

Study Guide 4

5. Lebesgue Integration: Further Topics.

5.1. Uniform Integrability and Tightness: A General Vitali Convergence Theorem. Section 5.1 notes

Supplement. Proofs of Theorems in Section 5.1. Beamer file of Section 5.1 proofs
Supplement. Printout of the Proofs of Theorems in Section 5.1. Print file of Section 5.1 proofs

5.2. Convergence in Measure. Section 5.2 notes

Supplement. Proofs of Theorems in Section 5.2. Beamer file of Section 5.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 5.2. Print file of Section 5.2 proofs

5.3. Characterization of Riemann and Lebesgue Integrability. Section 5.3 notes

Supplement. Proofs of Theorems in Section 5.3. Beamer file of Section 5.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 5.3. Print file of Section 5.3 proofs

Study Guide 5

6. Differentiation and Integration.

6.1. Continuity and Monotone Functions. Section 6.1 notes

Supplement. Proofs of Theorems in Section 6.1. Beamer file of Section 6.1 proofs
Supplement. Printout of the Proofs of Theorems in Section 6.1. Print file of Section 6.1 proofs

6.2. Differentiability of Monotone Functions: Lebesgue's Theorem. Section 6.2 notes

Supplement. Proofs of Theorems in Section 6.2. Beamer file of Section 6.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 6.2. Print file of Section 6.2 proofs

6.3. Functions of Bounded Variation: Jordan's Theorem. Section 6.3 notes

Supplement. Proofs of Theorems in Section 6.3. Beamer file of Section 6.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 6.3. Print file of Section 6.3 proofs

6.4. Absolutely Continuous Functions. Section 6.4 notes

Supplement. Proofs of Theorems in Section 6.4. Beamer file of Section 6.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 6.4. Print file of Section 6.4 proofs

6.5. Integrating Derivatives: Differentiating Indefinite Integrals. Section 6.5 notes (Contains a proof of the Fundamental Theorem of Lebesgue Calculus.)

Supplement. Proofs of Theorems in Section 6.5. Beamer file of Section 6.5 proofs
Supplement. Printout of the Proofs of Theorems in Section 6.5. Print file of Section 6.5 proofs

6.6. Convex Functions. Section 6.6 notes

Supplement. Proofs of Theorems in Section 6.6. Beamer file of Section 6.6 proofs
Supplement. Printout of the Proofs of Theorems in Section 6.6. Print file of Section 6.6 proofs

Study Guide 6.

7. The L^p Spaces: Completeness and Approximation.

7.1. Normed Linear Spaces. Section 7.1 notes
7.2. The Inequalities of Young, Holder, and Minkowski. Section 7.2 notes

Supplement. Proofs of Theorems in Section 7.2. Beamer file of Section 7.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 7.2. Print file of Section 7.2 proofs

7.3. L^p Is Complete: The Riesz-Fischer Theorem. Section 7.3 notes

Supplement. Proofs of Theorems in Section 7.3. Beamer file of Section 7.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 7.3. Print file of Section 7.3 proofs

7.4. Approximation and Separability. Section 7.4 notes

Supplement. Proofs of Theorems in Section 7.4. Beamer file of Section 7.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 7.4. Print file of Section 7.4 proofs

Study Guide 7.

8. The L^p Spaces: Duality and Weak Convergence.

8.1. The Riesz Representation for the Dual of L^p, 1 ≤ p < ∞. Section 8.1 notes

Supplement. Proofs of Theorems in Section 8.1. Beamer file of Section 8.1 proofs
Supplement. Printout of the Proofs of Theorems in Section 8.1. Print file of Section 8.1 proofs

8.2. Weak Sequential Convergence in L^p. Section 8.2 notes

Supplement. Proofs of Theorems in Section 8.2. Beamer file of Section 8.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 8.2. Print file of Section 8.2 proofs

Supplement. Proof of the Radon-Riesz Theorem for 1 < p < ∞. Radon-Riesz Theorem Supplement notes
8.3. Weak Sequential Compactness. Section 8.3 notes

Supplement. Proofs of Theorems in Section 8.3. Beamer file of Section 8.3 proofs
Supplement. Printout of the Proofs of Theorems in Section 8.3. Print file of Section 8.3 proofs

8.4. The Minimization of Convex Functionals.
Study Guide 8.

Chapter 5. Vector Spaces, Hilbert Spaces, and the L² Space.

5.1. Groups, Fields, and Vector Spaces (Section 5.1 from Real Analysis with an Introduction to Wavelets and Applications - There is a detailed discussion of Hamel and Schauder bases). Section 5.1 notes (Includes a proof that every vector space has a Hamel basis and that any two Hamel bases for a given vector space have the same cardinality.)

Supplement. Proofs of Theorems in Section 5.1. Beamer file of Section 5.1 proofs
Supplement. Printout of the Proofs of Theorems in Section 5.1. Print file of Section 5.1 proofs

5.2. Inner Product Spaces (Section 5.2 from Real Analysis with an Introduction to Wavelets and Applications). Section 5.2 notes

Supplement. Proofs of Theorems in Section 5.2. Beamer file of Section 5.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 5.2. Print file of Section 5.2 proofs

5.3. The Space L² (Section 5.3 from Real Analysis with an Introduction to Wavelets and Applications). Section 5.3 notes
5.4. Projections and Hilbert Space Isomorphisms (Section 5.4 from Real Analysis with an Introduction to Wavelets and Applications). Section 5.4 notes (Includes an example of a closed and bounded set that is not compact.)

Supplement. Proofs of Theorems in Section 5.4. Beamer file of Section 5.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 5.4. Print file of Section 5.4 proofs

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