The lecture topics have accompanying lecture summaries.
LEC # | TOPICS | KEY DATES |
---|---|---|
1 | Sets, ordered sets, countable sets | |
2 | Fields, ordered fields, least upper bounds, the real numbers | |
3 | The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, Cauchy-Schwarz | Problem set 1 due |
4 | Metric spaces, ball neighborhoods, open subsets | |
5 | Open subsets, limit points, closed subsets, dense subsets | Problem set 2 due |
6 | Compact subsets of metric spaces | |
7 | Limit points and compactness; compactness of closed bounded subsets in Euclidean space | Problem set 3 due |
Midterm | ||
8 | Convergent sequences in metric spaces; Cauchy sequences, completeness; Cauchy’s theorem | |
9 | Subsequential limits, lim sup and lim inf, series | Problem set 4 due |
10 | Absolute convergence, product of series | |
11 | Power series, convergence radius; the exponential function, sine and cosine | Problem set 5 due |
12 | Continuous maps between metric spaces; images of compact subsets; continuity of inverse maps | |
13 | Continuity of the exponential; the logarithm; Intermediate Value Theorem; uniform continuity | Problem set 6 due |
14 | Derivatives, the chain rule; Rolle’s theorem, Mean Value Theorem | |
15 | Derivative of inverse functions; higher derivatives, Taylor’s theorem | Problem set 7 due |
Midterm | ||
16 | Pointwise convergence, uniform convergence; Weierstrass criterion; continuity of uniform limits; application to power series | |
17 | Uniform convergence of derivatives | |
18 | Spaces of functions as metric spaces; beginning of the proof of the Stone-Weierstrass Theorem | Problem set 8 due |
19 | End of Stone-Weierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions) | |
20 | Riemann-Stjeltjes integral: definition, basic properties | |
21 | Riemann integrability of products; change of variables | Problem set 9 due |
22 | Fundamental theorem of calculus; back to power series: continuity, differentiability | |
23 | Review of exponential, log, sine, cosine; e^{it }= cos(t) + isin(t) | Problem set 10 due |
24 | Review of series, Fourier series | |
Final Exam |