Tuesday, August 19, 2014

Course Information

Time and Place: MWF 11:30am–12:20pm in REC 113

Instructor: Arshak Petrosyan

Office Hours: MWF 9:30-10:30, or by appointment, in MATH 610

Textbook: [R] W. Rudin, Principles of mathematical analysis, Third edition, McGraw-Hill, New York, 1976

Homework will be collected weekly on Wednesdays. The assignments will be posted on this website at least one week prior the due date.

Exams: There well be two midterm exams (evening exams) and a comprehensive final exam (covering all material). The exact time and place will be specified at least two weeks in advance.

Description: Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions, the Riemann-Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem, equicontinuity, and the Arzela-Ascoli theorem.

Chapter 1. The Real and Complex Number System
  • Real number system - (Emphasize inf, sup)
  • Extended real number system
  • Euclidean spaces
Chapter 2. Basic Topology
  • Finite, countable and uncountable sets
  • Metric spaces (Only a few special examples)
  • Compact sets
Chapter 3. Numerical Sequences and Series
  • Convergent sequences
  • Subsequences
  • Cauchy sequences
  • $\limsup x_n$ and $\liminf x_n$
  • Series
  • Series with many terms (comparison test)
  • Absolute and conditional convergence
  • Rearrangements
Chapter 4. Continuity
  • Limits of functions
  • Continuous functions
  • Continuity and compactness
  • Intermediate Value Theorem
Chapter 6. The Riemann-Stieltjes Integral
  • Definition and existence
  • Properties
  • Integration and differentiation
Chapter 7. Sequences and Series of Functions
  • Uniform convergence
  • Uniform convergence and continuity
  • Uniform convergence and integration
  • Uniform convergence and differentiation
  • Equicontinuous families of functions
  • Stone-Weierstrass Theorem
Optional Topics
  • Sets of Lebesgue measure zero
  • Characterization of Riemann integrable functions bounded and continuous a.e.
  • Differentiability a.e. of monotone functions