Calendar

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{{< sub “it” >}}= cos(t) + isin(t) Problem set 10 due
24 Review of series, Fourier series
Final Exam