Lecture Notes: Introduction to Real Analysis

Overview

• Focus on limits of functions, continuity, differentiability, sequences, and series of functions.
• Assumes knowledge in sequences and series of real numbers and basic point set topology.

Chapter 1: The Real Numbers

1.1 Completeness of R

• Completeness: Real numbers form a continuum with no gaps.
  • Suprema & Infima: Every nonempty set of real numbers that is bounded above has a supremum.
  • Cauchy Sequences: A sequence is Cauchy if it converges in R.

1.2 Open and Closed Sets

• Open Sets: A set G is open if every point is an interior point.
• Closed Sets: A set F is closed if its complement is open.

1.3 Accumulation and Isolated Points

• Accumulation Point: Every neighborhood contains infinite points from set A.
• Isolated Point: Exists in set A but no other points in some neighborhood.

1.4 Compact Sets

• Compactness: A subset is compact if it is closed and bounded (Heine-Borel).
  • Sequential Compactness: Every sequence has a convergent subsequence.

Chapter 2: Limits of Functions

2.1 Limits

• Definition: Limit exists if for every epsilon > 0, there is a delta > 0 such that the condition holds.

2.2 Limit Properties

• Left, Right, and Infinite Limits: Definitions for limits approaching from one side or towards infinity.
• Properties: Limits respect algebraic operations; unique if they exist.

Chapter 3: Continuous Functions

3.1 Continuity

• Definition: A function is continuous if it takes nearby values at nearby points.

3.2 Properties

• Algebraic Operations: Continuous functions preserve operations like sums and products.

3.3 Uniform Continuity

• Stronger form of continuity where delta is independent of x.

Chapter 4: Differentiable Functions

4.1 Derivative

• Definition: Differentiable if limit of the difference quotient exists.

4.2 Properties

• Differentiability Implies Continuity: If a function is differentiable, it is continuous.

4.3 Mean Value Theorem

• Theorem: Average rate of change equals instantaneous rate at some point.

Chapter 5: Sequences and Series of Functions

5.1 Pointwise Convergence

• Definition: Contributes to convergence at individual points.

5.2 Uniform Convergence

• Stronger Convergence: Uniform over the entire domain.

5.3 Properties

• Preservation: Uniform convergence preserves continuity and boundedness.

Chapter 6: Power Series

6.1 Introduction

• Definition: Series of the form Σaₙ(x-c)ⁿ.

6.2 Radius of Convergence

• Definition: Exists an R where series converges inside and diverges outside.

6.3 Differentiability

• Term-by-term Differentiation: Power series can be differentiated term by term.

Chapter 7: Metric Spaces

7.1 Metrics

• Definition: A set with a defined distance function satisfying specific properties.

7.2 Norms

• Normed Vector Space: Vector spaces with a norm defining lengths.

7.3 Open and Closed Sets

• Open sets contain a neighborhood of every point; closed sets are complements of open sets.

Conclusion

• Completeness and Compactness: Key concepts in understanding continuity, limits, and differentiability.