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 13-July-2016)
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 13-July-2016)

• Chapter 1: Basic Ideas (Updated 13-July-2016)
Basic set theory, notation, Schröder-Bernstein Theorem. Countability, uncountability and cardinal numbers.
• Chapter 2: The Real Numbers (Updated 13-July-2016)
Axioms of a complete ordered field and some consequences. The most basic properties of $$\mathbb{R}$$. Uncountability of $$\mathbb{R}$$.
• Chapter 3: Sequences (Updated 13-July-2016)
The most basic consequences of completeness.
• Chapter 4: Series (Updated 13-July-2016)
An application of sequences. Standard and some more advanced convergence tests.
• Chapter 5: The Topology of $$\mathbb{R}$$ (Updated 13-July-2016)
Various forms of completeness and compactness. Connectedness and relative topologies. Baire category. Measure zero.
• 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 rule.
• Chapter 8: Integration (Updated 13-July-2016)
Development of the Riemann-Darboux integral, Fundamental Theorem of Calculus.
• 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. Continuous function with divergent Fourier series. Gibbs phenomenon. Césaro convergence and pointwise convergence.

• Bibliography (Updated 13-July-2016)

• Index (Updated 13-July-2016)
This is a sparse index

Contact me at Lee Larson (llarson@louisville.edu)