COURSE DESCRIPTION
The objectives of this course are: (1) to provide you with a firm understanding of the basic ideas of mathematical analysis; (2) to improve your proof writing skills up to a professional level; (3) to improve problem solving skills needed for the CAAM analysis qualifier exam; and (4) to improve your mathematical communication skills. The specific analysis topics covered include Real numbers, completeness, sequences and convergence, compactness, continuity, the derivative, the Riemann integral, the fundamental theorem of calculus. vector spaces, dimension, linear maps, inner products and norms. See the syllabus.
Text Book: Undergraduate Analysis by Lang
Homeworks: This class has homework problems of two types: practice problems and portfolio problems. Practice problems are only graded for completion. Portfolio problems are problems that are similar to qualifier problems. They will be graded in a binary manner, taking into account the correctness, the elegance, and the exposition. Here are some examples of the quality of proofs expected. Portfolio problems can and should be revised until they are judged to be of professional quality. Two homework assignments will be pledged and will serve as simulated qualifier exams.
Classroom participation and in-class presentations: Each day of class, one or two students will present a theorem from that day’s content. The classroom participation grade will be based on the sincerity of your preparation for these presentations. Please follow my comments and suggested structure for the presentations.
Outside resources: You are not allowed to use the Problems and Solutions book accompanying Lang’s Undergraduate Analysis text for any of the homeworks.
Disabilities: Any student with a disability needing academic accommodations is requested to speak with me as soon as possible. All discussions will remain confidential. Students should also contact Disability Support Services in the Ley Student center.
SCHEDULE
| Event | Date | Related Documents |
|---|---|---|
| HW 1 | Sep 1 in class | Problems. Solutions |
| HW 2 | Sep 8 in class | Problems. Solutions |
| HW 3 | Sep 15 in class | Problems. Solutions |
| HW 4 | Sep 22 in class | Problems |
| Pledged HW 5 | Sep 29 in class | Problems. Solutions. Sample from 2014 |
| HW 6 | Oct 20 in class | Problems. |
| HW 7 | Oct 27 in class | Problems. |
| HW 8 | Nov 3 in class | Problems. |
| HW 9 | Nov 17 in class | Problems |
| HW 10 | Nov 24 in class | Problems |
| Pledged HW 11 | Dec 1 in class | Problems. Sample from 2014 |
Lecture Notes
| Day | Topics | Reference (Lang) | Class notes |
|---|---|---|---|
| Aug 25 | Real numbers. | I.1-I.4, II.1 | Summary. Notes |
| Aug 27 | Cauchy sequences, Bolzano-Weierstrass, liminf and limsup | II.1, II.2 | Summary. Notes |
| Sep 1 | Limits, continuity | II.2, II.4 | Summary. Notes |
| Sep 3 | Squeeze theorem, limits with infinity | II.2, II.3, III.1 | Summary. Notes |
| Sep 8 | Differentiability, Mean value theorem | III.2, III.3 | Summary. Notes |
| Sep 10 | Convex funcions, inverse function theorem | III.2, III.3 | Summary. Notes |
| Sep 15 | Riemann Integration | V.1-V.2 | Summary. Notes |
| Sep 17 | Riemann Integration | V.2 | Summary. Notes |
| Sep 22 | Taylor Series | VI.3 | Summary. Notes |
| Sep 24 | Slack | ||
| Sep 29 | Slack | ||
| Oct 1 | Norms | VI.1-VI.3 | Summary. Notes |
| Oct 6 | Inner products and norm equivalence | VI.1-VI.3 | Summary. Notes |
| Oct 8 | Complete normed vector spaces | VI.4 | Summary. Notes |
| Oct 15 | Open and closed sets | VI.5 | Summary. Notes |
| Oct 20 | Dimensionality of spaces | Notes by Symes | Summary. |
| Oct 22 | Limits in normed vector spaces and function spaces | VII.1, VII.2, VII.3 | Summary. Notes |
| Oct 27 | Equivalence relations | VII.4 | Summary. Notes |
| Oct 29 | Completion of spaces | VII.4 | Summary. Notes |
| Nov 3 | Sequential compactness, Compactness by open covers | VIII.1, VIII.2, VIII.4 | Summary. Notes |
| Nov 10 | Series | IX.1, IX.2, IX.3, IX.4, IX.5 | Summary. Notes |
| Nov 12 | Power series | IX.6-IX.7 | Summary. Notes |
| Nov 17 | Extension of linear functions | X.1, X.2, X.3 | Summary |
| Nov 19 | Integral via step functions | X.1-X.3 | Summary |
| Nov 24 | Measure and content | X.4 appendix | Summary |
| Dec 1 | Almost everywhere convergence | X.4 appendix | Summary |
| Dec 3 | Relation of derivatives and integrals | X.5-X.7 | Summary |