Lectures are MW 12:30PM - 1:45PM. Page numbers refer to Rudin, Principles of Mathematical Analysis, 3rd edition.
No class on Labor Day (Mon 9/7), Fall Break (Mon 10/12), and the Thanksgiving recess (Wed 11/25). The last day of classes is Mon 12/14.
| Week | Week of | Topic | Reading assignment | Homework |
|---|---|---|---|---|
| 1 | 8/31 | Ordered sets, fields, the real field (pg. 1-8) | The real field, the extended real number system, the complex field, Euclidean spaces (pg. 9-17) | |
| 2 | 9/7 | (no class Mon 9/7) The real field, the extended real number system, the complex field (pg. 9-16) | Finite, countable, and uncountable sets, metric spaces (pg. 24-36) | Homework 1 (due 9/17) |
| 3 | 9/14 | Euclidean spaces, finite, countable, and uncountable sets, metric spaces (pg. 16-17, 24-31) | Compact sets (pg. 36-40) | Homework 2 (due 9/24) |
| 4 | 9/21 | Metric spaces, compact sets, connected sets (pg. 32-40, 42-43) | Numerical sequences (pg. 47-58) | Homework 3 (due 10/1) |
| 5 | 9/28 | Convergent sequences, subsequences, Cauchy sequences, upper and lower limits (pg. 47-58) | Numerical series (pg. 58-69) | Homework 4 (due 10/8) |
| 6 | 10/5 | Numerical series (pg. 58-69) | Power series, summation by parts, absolute convergence, operations with series: addition, multiplication, rearrangements (pg. 69-78) | Homework 5 (due 10/15) |
| 7 | 10/12 | (no class Mon 10/12) Power series, summation by parts, absolute convergence, operations with series (pg. 69-78) | Homework 6 (due 10/22) | |
| 8 | 10/19 | Review session for the midterm exam | Limits of functions, continuous functions, continuity and compactness, continuity and connectedness (pg. 83-93) | |
| 9 | 10/26 | Midterm exam (Mon 10/26); limits of functions, continuous functions, continuity and compactness (pg. 83-90) | Discontinuities, monotonic functions, infinite limits and limits at infinity (pg. 94-98); the derivative of a real function, mean value theorems, the continuity of derivatives, L’Hospital’s rule (pg. 103-110) | |
| 10 | 11/2 | Continuity and compactness, continuity and connectedness, discontinuities, monotonic functions, infinite limits and limits at infinity (pg. 91-98); the derivative of a real function (pg. 103-105) | Derivatives of higher order, differentiation of vector-valued functions (pg. 110-113); definition and existence of the Riemann-Stieltjes integral (pg. 120-125) | Homework 7 (due 11/12) |
| 11 | 11/9 | Mean value theorems, the continuity of derivatives, derivatives of higher order, differentiation of vector-valued functions (pg. 107-113); definition and existence of the Riemann-Stieltjes integral (pg. 120-123) | Homework 8 (due 11/19); midterm corrections due | |
| 12 | 11/16 | Definition and existence of the Riemann-Stieltjes integral, properties (pg. 124-129) | Properties of the integral, integration and differentiation, integration of vector-valued functions, rectifiable curves (pg. 129-137); the main problem for sequences and series of functions (pg. 143-145) | Homework 9 (due 12/3) |
| 13 | 11/23 | (no class Wed 11/25) Properties of the integral, integration and differentiation, integration of vector-valued functions, rectifiable curves (pg. 129-137) | Uniform convergence, uniform convergence and continuity, uniform convergence and integration (pg. 146-152) | Homework 10 (due 12/10) |
| 14 | 11/30 | Uniform convergence, uniform convergence and continuity (pg. 143-150) | ||
| 15 | 12/7 | Uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, Weierstrass theorem (using Bernstein’s polynomials, slides by R. Kadison and Z. Liu) (pg. 150-153, 159-160) | Review for the final exam | Homework 11 (due 12/14) |
| 16 | 12/14 | Review session for the final exam (Mon 12/14, last day of classes) |