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