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. We'll try and find a time every week for a problem solving session
  • Evaluation. %15 short midterm exam (October 10), %25 long midterm exam  (November 7), 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
8/29
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

9
10/22
Continuous functions with real and complex values

10/24
Differentiability

10
10/29
The Riemann integral

11/31
Properties of the Riemann integral
11
11/5
The fundamental theorem of Calculus 
11/7 EXAM [Chapters 1 to 4]
LONG MIDTERM EXAM
12
11/12
The metric space of bounded complex-valued functions on a metric space
11/14
Uniform convergence (continuity, Weierstrass's M-test) + the metric space of continuous bounded functions
13
THANKSGIVING WEEK 
14
11/26
Uniform continuity and integration
11/28
The Arzela-Ascoli Theorem.

15
12/3
The Stone-Weierstrass Theorem

12/5
Proof of the Arzela-Ascoli Theorem + Final review


TBD
FINAL EXAM (Three hours.)