Math 403H     Fall 2021   Schedule

Lec.	Date	Chapter	Topic
1	8/23	2	Introduction.  Countable and uncountable sets.
2	8/25	2	Cantor set and Cantor function.
3	8/27	1, 2	Monotone functions.  More on Cantor function.
4	8/30	3	Metric spaces: definition and examples.
5	9/1	3	More on metrics.  Bounded sets.
6	9/3	3	Vector spaces review  pdf.   Normed vector spaces.
-	9/6	-	Labor Day - No Classes
7	9/8	3	Limits in metric spaces.
8	9/10	3	Cauchy sequences. "Coordinatewise" convergence.
9	9/13	4	Open sets.
10	9/15	4	Closed sets.
11	9/17	4	Interior and closure. Dense sets.
12	9/20	5	Continuous functions.
13	9/22	5	Continuous functions.
14	9/24	5	Homeomorphisms.
15	9/27	5	Isometries. Distance to a set.
16	9/29	5, 6	Continuous real-valued functions.  Connectedness.
17	10/1	6	Connectedness.
18	10/4		Problems   pdf
19	10/6		Exam 1  covering Chapters 2-5.
20	10/8	7	Review: Sequences in [a,b] and continuous functions on [a,b].     Boundedness and total boundedness.
21	10/11	7	Total boundedness.
22	10/13	7	Completeness.
23	10/15	7	Completions.
24	10/18	7	Nested closed sets. Contractions.
25	10/20	8	Compactness.
26	10/22	8	Compactness via open covers.
27	10/25	8	Continuous functions on compact metric spaces.
28	10/27	8	Uniform continuity.
29	10/29	8	Properties of uniformly continuous functions.
30	11/1	9	Hölder continuous functions.   Open dense sets.
31	11/3	9	Nowhere dense sets. The Baire Category Theorem.
32	11/5	9	Discontinuities.
33	11/8		Problems   pdf
34	11/10		Exam 2  covering Chapters 6-8.
35	11/12	10	Pointwise convergence and uniform convergence.
36	11/15	10	Consequences of uniform convergence.
37	11/17	10	Spaces of bounded and continuous functions. Weierstrass M-test.
38	11/19	10	Power series.
	11/21-27		Thanksgiving break.
39	11/29	11	Equicontinuity.  Equicontinuity and uniform convergence.
40	12/1	11	The Arzelà-Ascoli Theorem.
41	12/3	12	Subalgebras of C(X).
42	12/6	12	The Stone-Weierstrass Theorem.
43	12/8	12	Discussion and examples.
44	12/10		Problems   pdf

  Final Exam:  December 12,  2-5 p.m. via Canvas.