University of Pittsburgh Math 420 : Introduction to Analysis : Fall 2016

Math 420: Introduction to Analysis

Mon, Wed, Fri 2:00 - 2:50pm -- Benedum 1045

Instructor. Dr. Marta Lewicka (office hours in Thackeray 408, Mondays 3:00 - 4:00pm)
Teaching Assistant. Tian Jiang (recitations Tue 2:00-2:50pm -- Allen 105)

Textbook. Jiri Lebl: Basic Analysis. Custom Pitt edition.
Prerequisites. The course concerns the One-Variable Calculus. Prerequisite is Math 413: Introduction to Theoretical Mathematics. If you do not feel comfortable with the prerequisite material, please contact the instructor in the beginning of the course.
Grades. Your final grade depends on your performance on the final exam as well as on your total grade. Grades will be based on homework (35%), two midterms (15% + 15%) and the final (35%). There will be no make up midterm exams. If you miss the midterm exam for a *documented* medical reason, your grade on it will be the prorated grade of your final exam. Incompletes will almost never be given, and only for cases of extreme personal tragedy.
Homework. Homework will be assigned each Monday (starting in the second week of the semester), and it will be due on Tuesday the following week during the recitations. Late homework will not be accepted. The solution of each exercise will be evaluated in the scale 0-5 points, taking into account the correctness, clarity and neatness of presentation.
Core topics.
1. The Bolzano-Weierstrass Theorem; Cauchy sequences; Cauchy completeness of the real numbers.
2. Real-valued functions on an interval: limits and continuity.
3. Intermediate Value Theorem; Max-Min Theorem.
4. Uniform continuity; continuous functions on a closed and bounded interval are uniformly continuous.
5. Differentiable functions.
6. Interior Extremum Theorem, Rolle's Theorem, Mean Value Theorem.
7. Taylor's Theorem and Taylor Series.
8. The Riemann Integral on a closed and bounded interval.
9. The Fundamental Theorem of Calculus.
10. Definition and examples of pointwise and uniformly convergent sequences of functions.
11. Continuity of uniform limits of continuous functions.
12. Interchange of uniform limits and integration.
13. Interchange of limits with differentiation.
14. The M-test for uniform convergence of series.
15. Application to power series.

Calendar.

7 Jan (Mon): First Class
21 Jan (Mon): Martin Luther King's day - no classes
Feb 15 (Fri): First Midterm
11, 13, 15 Mar (Mon, Wed, Fri): Spring Break - no classes
???: Second Midterm
19 Apr (Fri): Last Class
Final Exam: Monday April 22, 10am-11:50am