Readings

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  

Course Info