Lecture Notes: An Introduction to Real Analysis

John K. Hunter

Department of Mathematics, University of California at Davis

• The lecture provides notes on introductory real analysis covering:
  • Limits of functions
  • Continuity
  • Differentiability
  • Sequences and series of functions
  • Not covering Riemann integration
• Assumes a background in sequences, series of real numbers, and elementary topology of real numbers.

Contents

Chapter 1: The Real Numbers

1. Completeness of R

   • R forms a complete continuum.
   • Definitions of Suprema and Infima.

2. Open Sets

   • Definition and properties of open sets and neighborhoods.

3. Closed Sets

   • Definition and characterizations (e.g. sequentially closed).

4. Accumulation Points and Isolated Points

   • Definition and examples.

5. Compact Sets

   • Sequential compactness and Heine-Borel theorem.

Chapter 2: Limits of Functions

1. Limits

   • Definitions and properties.

2. Left, Right, and Infinite Limits

   • Definitions and examples.

3. Properties of Limits

   • Uniqueness, boundedness, algebraic properties.

Chapter 3: Continuous Functions

1. Continuity

   • Definitions and examples.

2. Properties of Continuous Functions

   • Addition, multiplication, and composition.

3. Uniform Continuity

   • Definitions and distinctions from ordinary continuity.

4. Continuous Functions and Open Sets

   • Topological properties.

5. Continuous Functions on Compact Sets

   • Boundedness, extreme values, and uniform continuity.

6. Intermediate Value Theorem

   • Statement and implications.

7. Monotonic Functions

   • Definitions and properties.

Chapter 4: Differentiable Functions

1. The Derivative

   • Definitions and examples.

2. Properties of the Derivative

   • Continuity, linearity, product, and quotient rules.

3. Extreme Values

   • Fermat's theorem and critical points.

4. Mean Value Theorem

   • Rolle's theorem and implications.

5. Taylor's Theorem

   • Taylor polynomials and remainder.

Chapter 5: Sequences and Series of Functions

1. Pointwise Convergence

   • Definitions and examples.

2. Uniform Convergence

   • Definitions and examples.

3. Cauchy Condition for Uniform Convergence

   • Necessary and sufficient conditions.

4. Properties of Uniform Convergence

   • Boundedness, continuity, and differentiability.

5. Series

   • Definitions and convergence tests.

6. Weierstrass M-Test

   • Uniform convergence criterion.

7. Sup-norm and Spaces of Continuous Functions

   • Definitions and properties.

Chapter 6: Power Series

1. Introduction

   • Definition and basic properties.

2. Radius of Convergence

   • Definitions and methods to find it.

3. Examples of Power Series

   • Common series and their radii of convergence.

4. Differentiation of Power Series

   • Term-by-term differentiation.

5. The Exponential Function

   • Definition via power series.

6. Taylor's Theorem and Power Series

   • Smooth vs. analytic functions.

Chapter 7: Metric Spaces

1. Metrics

   • Definition and examples.

2. Norms

   • Definition and examples.

3. Sets

   • Open, closed, and compact sets.

4. Sequences

   • Convergence, Cauchy sequences, and completeness.

5. Continuous Functions

   • Definitions and properties.

6. Appendix: The Minkowski Inequality

   • Proofs and application in analysis.

• These notes provide a comprehensive overview of foundational concepts in real analysis, suitable for students with a background in sequences and topology.