** Class: ** TuTh, 12:30-2:00 PM, Etcheverry 3107

** Instructor: ** Ved V. Datar

** Email: ** vv *lastname* at math.berkeley.edu, no spaces

** Office: ** 1067 Evans Hall

** Office hours: **M 2-3:30PM, Th 2-3PM, or by appointment (if you
have another class during office hours).

** GSI: **Edward Scerbo

** GSI Office hours:** MW 2-4, TuTh
4-6, F 1030AM-1230

** GSI Office hours location:** MWF - Evans 732, Tu - Evans
762, Th - Evans 732 or 748 (you need to check both rooms)

** Text: **Walter Rudin, *Principles of mathematical
analysis*, (any edition)

** Suplementary reading: ** Charles Pugh, *Real mathematical analysis*.

Homeworks - 20%, Midterms - 40%, Final - 40%

There will be 12 homeworks. The best ten will be counted towards
the grade. ** There is no late submission of homeworks. **

There will be two midterms of 40 points each and a final exam of
80 points. ** There will be no make-up exams.** The midterm score
will be calculated by the formula -

If you are a student with disability (with or without a document from the Disabled Students' Program) and require special accommodations of any kind, please e-mail the instructor as soon as possible, and no later than September 04th.

If you are representing the university on some official duties (say if you are an athlete or in a band), and if there is a conflict with any of the mid-terms, please let the instructor know before September 04th.

To do well in the course, it is important to be able to understand as well as write proofs. It might be useful to read Prof. Hutching's article on mathematical reasoning.

Most of our proofs will use standard set theoretc notation. Here are some notes from Prof. Bergman on set theory.

We will also often use mathematical induction. Here are some notes on that by Prof. Bergman.

Practice problems for first mid-term, and the solutions

First midterm , and Solutions

Practice problems for second mid-term, and the solutions

The page numbers refer to the third edition of Rudin's book.

Number | Date | Topic | Reading | Homework | Notes |

1 | Th 08/24 | introduction, cardinality of sets, review of rationals and irrationals | 24-29 | A1 (due 09/05), Solutions. | Cardinality |

2 | Tu 08/29 | ordered fields, supremum and infimum, construction of reals | 1-11 | Real and complex numbers | |

3 | Th 08/31 | complex numbers, Euclidean space, metric spaces - definition and examples | 12-17, 30-31 | ||

4 | Tu 09/05 | open and closed sets, limit points | 32-36 | A2 (due 09/12), Solutions. | Metric Spaces - Opena and closed sets |

5 | Th 09/07 | metric subspaces, compact sets | 36-38 | ||

6 | Tu 09/12 | compact sets in R^n, connected set definition | 39-40, 42 | A3 (due 09/19) Solutions | Compact and connected sets |

7 | Th 09/14 | connected sets (cont.), connected sets in R, sequences | 42, 47-51 | ||

8 | Tu 09/19 | Cauchy criteria, completeness | 52-54 | A4 (due 10/03) Solutions | Sequences |

9 | Th 09/21 | sequences of real numbers, examples limsup, liminf | 55-58 | ||

10 | Tu 09/26 | midterm-1 | 53-55 | ||

11 | Th 09/28 | limits (including infinite limits) and continuity, continuity and open/closed sets | 83-88 | Limits and Continuity | |

12 | Tu 10/03 | continuity and compact sets, extremum value theorem uniform continuity | 89-92 | A5 (due 10/10) Solutions | |

13 | Th 10/05 | continuity and connected sets, itermediate value theorem, discontinuities | 93-95 | ||

14 | Tu 10/10 | differentiability, examples, chain rule | 103-106 | A6 (due 10/17) Solutions | Differentiation |

15 | Th 10/12 | mean value theorems, applications to optimization, Taylor's theorem | 107-108, 111 | ||

16 | Tu 10/17 | definition and existence of intergral, properties of integral | 120-124 | A7 (due 10/24) Solutions | |

17 | Th 10/19 | criteria for integrability, applications | 124-127 | Integration | |

18 | Tu 10/24 | integration and differentiation | 133-134 | A8 (due 10/31) Solutions | |

19 | Th 10/26 | sequences of functions, examples, uniform convergence definition, example | 143-147 | Uniform Convergence | |

20 | Tu 10/31 | Cauchy criteria, relation with continuity and integration | 148-152 | A9 (due 11/07) Solutions | |

21 | Th 11/02 | uniform convergence and differentiation, Stone-Weierstrass theorem | 152-153, 159-160 | ||

22 | Tu 11/07 | Equicontinuity, Ascoli-Arzela | 155-158 | ||

23 | Th 11/09 | Ascoli-Arzela (cont.) | 155-158 | ||

24 | Tu 11/14 | midterm-2 |
105-107 | ||

25 | Th 11/16 | Arzela-Ascoli (cont.) | 155-158 | A10 (due 11/21) | |

26 | Tu 11/21 | root and ratio tests, infinite series of functions, Weierstrass' M-test | 66-68, 148 | A11 (due 11/28) | |

27 | Tu 11/28 | power series, fundamental theorem of power series | 172-178 | A12 (due 12/05) | |

28 | Th 11/30 | power series for exp, log, sine and cosine | 178-184 | ||

Tu 12/05 | RRR | 128-132 | |||

Th 12/07 | RRR |

Last modified: Thu Nov 16 12:15:07 PST 2017