Time and Place: MWF 12:30pm–1:20pm, UNIV 217

Instructor: Arshak Petrosyan

Office Hours: MWF 10:30am-11:30m, or by appointment, in MATH 610

Course Description: Credit Hours: 3.00. 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.

Textbook:

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

Course Outline:

  • 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

Homework will be collected weekly on Wednesdays (with some exceptions), at the beginning of class. No late homeworks will be accepted, however, the lowest homework score will be dropped. The assignments will be posted here at least one week prior to the due date.

Exams: There will be two midterm exams (evening exams) and a final exam . The exact information will be posted in the Exams page.

Grading: Your final grade will be computed by the scheme

Final Score = (1/3)FE + (7/30)ME1 + (7/30)ME2 + (1/5)HW,

where where FE, MEi, HW are the scores (in %) for Final Exam, Midterm i, Homework.

Note: If you perform better than average on both midterm exams, you will be given an option of not taking the final exam and your score will be computed by an alternative scheme (to be specified towards the end of the course).

Academic Integrity: As a reminder, all students must comply with Purdue’s policy for academic integrity:

https://www.purdue.edu/odos/osrr/academic-integrity-brochure/

Students with Disabilities: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.

In this mathematics course accommodations are managed between the instructor, student and DRC Testing Center.

You should see your instructor before or after class or during office hours to share your Accommodation Memorandum for the current semester and discuss your accommodations as soon as possible.

Emergencies: In the event of a major campus emergency, course requirements, deadlines and grading percentages are subject to changes that may be necessitated by a revised semester calendar or other circumstances beyond the instructor’s control. Relevant changes to this course will be posted onto the course website or can be obtained by contacting the instructor via email or phone. You are expected to read your @purdue.edu email on a frequent basis.