Real Analysis

Lecture Summaries

The following table contains summaries for each lecture topic listed.

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