Introduction to Real Analysis

This is a collection of lecture notes I’ve used several times in the two-semester senior/graduate-level real analysis course at the University of Louisville. They are an ongoing project and are often updated. They are here for the use of anyone interested in such material. In return, I only ask that you tell me of mistakes, make suggestions for improvements and, when sharing with others, please give me credit or blame.

I’m very interested in feedback of any type, so don’t be shy about contacting me!

• All chapters as one file (Updated 10-November-2017)
  • This file has the advantage that it is filled with hyperlinks, making it easier to track references.

Following are the chapters broken out into individual files. The advantage of these files is that when small updates are done, they show up here first. The disadvantage is they are not hyperlinked.

• Contents (Updated 23-June-2017)
  
  
• Chapter 1: Basic Ideas (Updated 23-June-2017)
  • Basic set theory
  • notation
  • Schröder-Bernstein Theorem
  • countability, uncountability
  • cardinal numbers
• Chapter 2: The Real Numbers (Updated 23-June-2017)
  • axioms of a complete ordered field
  • basic properties of \(\mathbb{R}\)
  • uncountability of \(\mathbb{R}\)
• Chapter 3: Sequences (Updated 23-June-2017)
  
  • monotone sequences
  • Nested Interval Theorem
  • Cauchy sequences
  • contractions
• Chapter 4: Series (Updated 23-June-2017)
  • positive series
  • convergence tests
  • absolute convergence
  • conditional convergence
  • alternating series
  • Riemann’s rearrangement theorem
• Chapter 5: The Topology of \(\mathbb{R}\) (Updated 23-June-2017)
  • open and closed sets
  • limit points
  • relative topologies
  • compactness
  • Baire Category Theorem
  • measure zero sets
  • Cantor middle-thirds set
• Chapter 6: Limits of Functions (Updated 13-July-2016)
  • limits
  • unilateral limits
  • continuity
  • uniform continuity
• Chapter 7: Differentiation (Updated 13-July-2016)
  • differentiation of functions
  • Darboux property
  • Mean Value Theorem
  • Taylor’s Theorem
  • l’Hôspital’s rules
• Chapter 8: Integration (Updated 13-July-2016)
  • Riemann-Darboux integral
  • Fundamental Theorem of Calculus.
  • change of variables
• Chapter 9: Sequences of Functions (Updated 13-July-2016)
  • pointwise convergence
  • uniform convergence and its relation to continuity, integration and differentiation
  • Weierstrass Approximation Theorem
  • power series.
• Chapter 10: Fourier Series (Updated 13-July-2016)
  • Dirichlet and Fejér kernels
  • Césaro convergence
  • pointwise convergence.
  • continuous function with divergent Fourier series
  • Gibbs phenomenon
• Bibliography (Updated 13-July-2016)
• Index (Updated 13-July-2016)
  

Contact me at Lee Larson (lee.larson@louisville.edu)