Math 104 : Real Analysis, Spring 2017

Basic Information

Class: MWF, 12:00-1:00 PM, 310 Hearst Mining

Instructor: Ved V. Datar

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

Office: 1067 Evans Hall

Office hours: MW, 11:00AM-12:00PM

GSI: Harrison Chen, Office hours - Evans 959, Tuesday 4-7, Wednesday 3-7, Thursday 4-7. Check his webpage for last minute cancellations.

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 disabled student (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 January 31st.

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 January 31st.

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. Here are the solutions

First midterm

Practice problems for the second mid-term.

Second midterm

Practice problems for the final

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 W 1/18 introduction, review of integers and rational number, ordered sets 1-5 A1 (due 01/30).
2 F 1/20 field axioms, construction of reals 5-10
3 M 1/23 complex numbers, Euclidean spaces, cardinality 12-17, 24-25 Lectures 1-3
4 W 1/25 cardinality (cont.), metric spaces 26-31
5 F 1/27 metric spaces (cont.) 16-18
6 M 1/30 metric spaces (cont.), compact sets 32-36 A2 (due 02/08).
7 W 2/01 compact sets 37-40
8 F 2/03 compact sets (cont.), connected sets 42 Lectures 4-8 (Cardinality and metric spaces)
9 M 2/06 convergent sequences and subsequences 47-52
10 W 2/08 Cauchy sequences and completeness 53-55 A3 (due 02/15).
11 F 2/10 monotonic sequences, limsup and liminf, examples 56-58 Sequences
12 M 2/13 infinite series, divergence test, comparison test59-60
13 W 2/15 p-series, geometric series, the number e 61-65 A4 (due 02/22)
14 F 2/17 root and ratio tests 65-69 Infinite Series - 1
M 2/20 holiday
15 W 2/22 midterm-1
16 F 2/24 summation by parts, Dirichlet's test, absolute and conditional convergence 70-72 Infinite Series - 2
17 M 2/27 rearrangements 75-78 A5 (due 03/08)
18 W 3/01 limits and continuity 83-86 Limits and Continuity (complete set)
19 F 3/03 equivalent formulations of continuity, examples 87-89, 98
20 M 3/06 continuity and compact sets, uniform continuity 90-92
21 W 3/08 continuity and connected sets, intermediate value theorem 93-97 A6 (due 03/15)
22 F 3/10 differentiability, basic properties 103,105
23 M 3/13 chain rule, mean value theorems 105-107 Derivatives
24 W 3/15 mean value theorems (cont.), L'Hopital's theorem 108-109 A7 (due 03/22)
25 F 3/17 derivatives of higher order, Taylor's theorem 110-111 Exponentials and Log
26 M 3/20 Riemann integral 120-121
27 W 3/22 sufficiency conditions for integrability, examples 124-126
28 F 3/24 properties of integral, change of variables 128-132 A8 (due 04/05)
M 3/27 spring recess
W 3/29 spring recess 91
F 3/31 spring recess
29 M 4/03 fundamental theorem of calculus, integration by parts Integration
30 W 4/05 midterm-2 133-134
31 F 4/07 improper integrals, integral test for series A9 (due 04/12)
32 M 4/10 sequences of functions, uniform convergence 143-148 Uniform Convergence
33 W 4/12 uniform convergence and continuity and integrals149-152 A10 (due 04/19)
34 F 4/14 uniform convergence and differentiation 152-154
35 M 4/17 equicontinuity and Ascoli-Arzela 155-158Ascoli-Arzela, Stone-Weierstrass
36 W 4/19 Stone-Weierstrass 159-161 A11 (due 04/26)
37 F 4/21 power series 69-70, 172-178Power Series
38 M 4/24 power series (cont.) 69-70, 172-178
39 W 4/26 power series (cont.)69-70, 172-178 A12 (due 05/03)
40 F 4/28 exp., log, trig functions 179-184
41 M 5/01 RRR
42 W 5/03 RRR
43 F 5/05 RRR

Last modified: Wed Aug 23 09:07:00 PDT 2017