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 improvement and, when sharing with others, please give me credit or blame.
- Chapter 1: Basic Ideas (Updated 7-May-10)
Basic set theory, notation, Schröder-Bernstein Theorem. Countability, uncountability and cardinal numbers.
- Chapter 2: The Real Numbers (Updated 7-May-10)
Axioms of a complete ordered field and some consequences.
- Chapter 3: Sequences (Updated 7-May-10)
The most basic consequences of completeness.
- Chapter 4: Series
An application of sequences. Standard and some more advanced convergence tests.
- Chapter 5: The Topology of ℝ
Various forms of completeness and compactness. Connectedness and relative topologies.
- Chapter 6: Limits of Functions
Limits, unilateral limits, continuity, uniform continuity.
- Chapter 7: Differentiation
Differentiation of functions, Darboux property, Mean Value Theorem, Taylor’s Theorem, l’Hôspital’s rule.
- Chapter 8: Integration
Development of the Riemann-Darboux integral, Fundamental Theorem of Calculus.
- Chapter 9: Sequences of Functions
Pointwise convergence. Uniform convergence and its relation to continuity, integration and differentiation. Weierstrass approximation theorem. Power series.
- Chapter 10: Fourier Series
Dirichlet and Fejér kernels. Césaro convergence and pointwise convergence.
- Bibliography (Updated 7-May-10)
Contact me at Lee Larson (firstname.lastname@example.org)