Real Analysis

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 #: 1 · TOPICS: Sets, ordered sets, countable sets · READINGS: pp. 3–6, and 24–30; Theorem 2.14 makes good reading even though it wasn’t covered in class.

LEC #: 2 · TOPICS: Fields, ordered fields, least upper bounds, the real numbers · READINGS: pp. 5–11

LEC #: 3 · TOPICS: The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, Cauchy-Schwarz · READINGS: pp. 38–9 (Theorem 2.38 only), 41 (Corollary to Theorem 2.43 only), and 12–6 Independent Reading: pp. 16–7 (Euclidean spaces)

LEC #: 4 · TOPICS: Metric spaces, ball neighborhoods, open subsets · READINGS: pp. 30–4

LEC #: 5 · TOPICS: Open subsets, limit points, closed subsets, dense subsets · READINGS: pp. 34–46

LEC #: 6 · TOPICS: Compact subsets of metric spaces · READINGS: pp. 36–8, and Problem 26 from p. 45

LEC #: 7 · TOPICS: Limit points and compactness; compactness of closed bounded subsets in Euclidean space · READINGS: pp. 38–40

LEC #: 8 · TOPICS: Convergent sequences in metric spaces; Cauchy sequences, completeness; Cauchy’s theorem · READINGS: pp. 47–9, and 51 (bottom of the page) –55

LEC #: 9 · TOPICS: Subsequential limits, lim sup and lim inf, series · READINGS: pp. 55–7, and 61–3

LEC #: 10 · TOPICS: Absolute convergence, product of series · READINGS: pp. 71–4, and 78

LEC #: 11 · TOPICS: Power series, convergence radius; the exponential function, sine and cosine · READINGS: pp. 69–70 plus some additional material from Chapter 8 (notably pages 178–80 on the exponential series, but covered only partially).

LEC #: 12 · TOPICS: Continuous maps between metric spaces; images of compact subsets; continuity of inverse maps · READINGS: pp. 83–9

LEC #: 13 · TOPICS: Continuity of the exponential; the logarithm; Intermediate Value Theorem; uniform continuity · READINGS: pp. 90–6

LEC #: 14 · TOPICS: Derivatives, the chain rule; Rolle’s theorem, Mean Value Theorem · READINGS: pp. 103–8

LEC #: 15 · TOPICS: Derivative of inverse functions; higher derivatives, Taylor’s theorem · READINGS: pp. 109–11

LEC #: 16 · TOPICS: Pointwise convergence, uniform convergence; Weierstrass criterion; continuity of uniform limits; application to power series · READINGS: pp. 143–51, and also part of Theorem 8.1 (p. 173)

LEC #: 17 · TOPICS: Uniform convergence of derivatives · READINGS: pp. 152–4; we also proved a weaker version of Theorem 7.25, just for functions of real numbers.

LEC #: 18 · TOPICS: Spaces of functions as metric spaces; beginning of the proof of the Stone-Weierstrass Theorem · READINGS: Definition 7.14 (the class has a bit more than that), Theorems 7.31 and 7.32

LEC #: 19 · TOPICS: End of Stone-Weierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions) · READINGS: Problem 23 on p. 169; the rest of Theorem 7.32

LEC #: 20 · TOPICS: Riemann-Stjeltjes integral: definition, basic properties · READINGS: pp. 120–30 (ending with a version of Theorem 6.15)

LEC #: 21 · TOPICS: Riemann integrability of products; change of variables · READINGS: Theorems 6.13, 6.17, 6.19

LEC #: 22 · TOPICS: Fundamental theorem of calculus; back to power series: continuity, differentiability · READINGS: pp. 133–4, and the parts of Theorem 8.1 which we didn’t do before.

LEC #: 23 · TOPICS: Review of exponential, log, sine, cosine; e it = cos(t) + isin(t) · READINGS: pp. 178–84

Apostol, Tom M. Mathematical Analysis . 2nd ed. Pearson Education, 1974. ISBN: 9780201002881.

Spivak, Michael. Calculus . 4th ed. Publish or Perish, 2008. ISBN: 9780914098911.