Fundamentals of Real Analysis

Course description. Rigorous introduction to classical real analysis. Brief review of real numbers. Full discussion of the basic topology of metric spaces, continuity and compactness. Differential analysis of functions of one real variable. Sequences and series of functions.

  • MW 3:35pm-4:50pm @ EWG209
  • Instructor. Francisco-Javier (Pancho) Sayas. My personal website is here and my group's site is here.
  • Textbook. Walter Rudin, Principles of Mathematical Analysis, Third edition.
  • Office hours. By appointment. Email me or talk to me in class to get an appointment. I will not reteach lectures during office hours. I will not solve your homework problems either. The goal of homework problems is that you learn how to solve them. The solution is less relevant than you ability to solve the problem.
  • Problem solving sessions. MW 5pm-6pm in EWG209
  • Evaluation. %15 short midterm exam (October 5), %25 long midterm exam  (November 2), 30% final exam (date TBD), 30% pop quizzes (average of all grades, eliminating the worst one)
  • No graded homework

Schedule

Week
Date
Topic
Additional info
1
8/31
Q as an ordered field and what's missing in it
Worksheets: (1) logic, (2) maps
2
9/7
R and its extensions
Problems Chapter 1
3
9/12
The metric structure of R^k
Worksheet: (3) countable sets
9/14
Metric spaces: neighborhoods, open sets
Problems Chapter 2
4
9/19
Metric spaces: limit points, closed sets, closure
Worksheet: (4) subsets
9/21
Compact sets
Problems Chapter 2.B
5
9/26
Compact sets in R^k
Worksheet: (5) connected sets
9/28
Sequences in a metric space
Problems Chapter 2.C
6
10/3
Cauchy and convergent sequences

10/5
EXAM
SHORT MIDTERM EXAM
7
10/10
Sequences in R^k, R, and C Problems Chapter 3.A
10/12
Lim-sup, lim-inf, and a taste of series

8
10/17
Continuity

10/19
Continuity, compactness and connectedness
Problems Chapter 3.B
9
10/24
Continuous functions with real and complex values
Problems Chapter 4
10/26
Differentiability (EXTRA TIME)

10
10/31
The Riemann Integral
Problems Chapter 5
11/2
[No class today]

11
11/7
[No class today]

11/9
EXAM
LONG MIDTERM EXAM
12
11/14
Properties of the Riemann integral
Problems Chapter 6
11/16
The fundamental theorem of Calculus + The metric space of bounded complex-valued functions on a metric space

13
THANKSGIVING WEEK   (Worksheet on Series)
14
11/28
Uniform convergence (continuity, Weierstrass's M-test) + the metric space of continuous bounded functions

11/30
Uniform continuity and integration + The Arzela-Ascoli Theorem. (EXTRA TIME)

15
12/5
The Stone-Weierstrass Theorem
Problems Chapter 7
12/7
Proof of the Arzela-Ascoli Theorem + Final review


12/12
FINAL EXAM (Three hours, starting at 3:30pm. Usual place.)