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. Fridays at 10am in EWG336
  • Evaluation. %15 short midterm exam (October 10), %25 long midterm exam  (November 7), 30% final exam (December 13), 30% pop quizzes (average of all grades, eliminating the worst one)
  • No graded homework

Schedule

Week
Date
Topic
Additional info
1
8/29
Q as an ordered field and what's missing in it
Worksheets: (1) logic, (2) countable sets
2
9/5
R and the Archimedean property, with an intro to C
Chapter 1
3
9/10
The metric structure of R^k
Worksheet (3) Sets
9/12
Metric spaces: neighborhoods, interior, open sets
Chapter 2 (part 1/3)
4
9/17
Metric spaces: closure, closed sets, limit points

9/19
Compact sets
Chapter 2 (part 2/3)
5
9/24
Compact sets in R^k
Chapter 2 (part 3/3)
9/26
Sequences in a metric space

6
10/1
Cauchy and convergent sequences
Chapter 3 (part 1/2)
10/3
Sequences in R^k, R, and C
7
10/8
Lim-sup, lim-inf, and a taste of series
10/10
EXAM [Chapters 1 and 2]
SHORT MIDTERM EXAM
8
10/15
Continuity
Chapter 3 (part 2/2)
10/17
Continuity, compactness, and connectedness
Connected sets
9
10/22
Continuous functions with real and complex values
Chapter 4
10/24
Differentiability

10
10/29
Differentiability (2): Mean value theorems and Taylor's theorem
Chapter 5
11/31
The Riemann integral
11
11/5
Properties of the Riemann integral 
11/7 EXAM [Chapters 1 to 5 - all the way to differentiability]
LONG MIDTERM EXAM
12
11/12
The fundamental theorem of Calculus Chapter 6
11/14
Uniform convergence

13
THANKSGIVING WEEK  (Work of this problem sheet: Series)
14
11/26
 NO CLASS TODAY

11/28
The space of bounded complex-valued functions
Chapter 7
15
12/3
The Arzela-Ascoli Theorem

12/5
The Stone-Weierstrass Theorem

12/13
FINAL EXAM [EVERYTHING] EWG209, starting at 3:30pm. Three hours long
Two exam problems