## 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.

Syllabus:
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