Math 104 : Real Analysis, Fall 2017

Basic Information

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.

Grade Distribution

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 -

MT score = MT1 + MT2 + F/2 - min(MT1,MT2,F/2).
To pass the class, you have to take the final exam.

Special Announcements

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.

Writing Proofs

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.

Announcements

Practice problems for first mid-term, and the solutions

First midterm , and Solutions

Practice problems for second mid-term, and the solutions

Tentative schedule, homeworks and lecture notes

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 continuity89-92 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 124-127Integration
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 143-147Uniform 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 172-178A12 (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