The readings are assigned in the textbook for this course:
Rudin, Walter. Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics). 3rd ed. McGraw-Hill, 1976. ISBN: 9780070542358.
| LEC # | TOPICS | READINGS |
|---|---|---|
| 1 | Sets, ordered sets, countable sets |
pp. 3–6, and 24–30; Theorem 2.14 makes good reading even though it wasn’t covered in class. |
| 2 | Fields, ordered fields, least upper bounds, the real numbers | pp. 5–11 |
| 3 | The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, Cauchy-Schwarz |
pp. 38–9 (Theorem 2.38 only), 41 (Corollary to Theorem 2.43 only), and 12–6 Independent Reading: pp. 16–7 (Euclidean spaces) |
| 4 | Metric spaces, ball neighborhoods, open subsets | pp. 30–4 |
| 5 | Open subsets, limit points, closed subsets, dense subsets | pp. 34–46 |
| 6 | Compact subsets of metric spaces | pp. 36–8, and Problem 26 from p. 45 |
| 7 | Limit points and compactness; compactness of closed bounded subsets in Euclidean space | pp. 38–40 |
| 8 | Convergent sequences in metric spaces; Cauchy sequences, completeness; Cauchy’s theorem | pp. 47–9, and 51 (bottom of the page) –55 |
| 9 | Subsequential limits, lim sup and lim inf, series | pp. 55–7, and 61–3 |
| 10 | Absolute convergence, product of series | pp. 71–4, and 78 |
| 11 | Power series, convergence radius; the exponential function, sine and cosine | pp. 69–70 plus some additional material from Chapter 8 (notably pages 178–80 on the exponential series, but covered only partially). |
| 12 | Continuous maps between metric spaces; images of compact subsets; continuity of inverse maps | pp. 83–9 |
| 13 | Continuity of the exponential; the logarithm; Intermediate Value Theorem; uniform continuity | pp. 90–6 |
| 14 | Derivatives, the chain rule; Rolle’s theorem, Mean Value Theorem | pp. 103–8 |
| 15 | Derivative of inverse functions; higher derivatives, Taylor’s theorem | pp. 109–11 |
| 16 | Pointwise convergence, uniform convergence; Weierstrass criterion; continuity of uniform limits; application to power series | pp. 143–51, and also part of Theorem 8.1 (p. 173) |
| 17 | Uniform convergence of derivatives | pp. 152–4; we also proved a weaker version of Theorem 7.25, just for functions of real numbers. |
| 18 | Spaces of functions as metric spaces; beginning of the proof of the Stone-Weierstrass Theorem | Definition 7.14 (the class has a bit more than that), Theorems 7.31 and 7.32 |
| 19 | End of Stone-Weierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions) | Problem 23 on p. 169; the rest of Theorem 7.32 |
| 20 | Riemann-Stjeltjes integral: definition, basic properties | pp. 120–30 (ending with a version of Theorem 6.15) |
| 21 | Riemann integrability of products; change of variables | Theorems 6.13, 6.17, 6.19 |
| 22 | Fundamental theorem of calculus; back to power series: continuity, differentiability | pp. 133–4, and the parts of Theorem 8.1 which we didn’t do before. |
| 23 | Review of exponential, log, sine, cosine; e it = cos(t) + isin(t) | pp. 178–84 |
| 24 | Review of series, Fourier series |
