Spring 2015 MTH 436 Mathematical Analysis
and Topology II
MW 3-4:15, F 3-3:50, Lippitt 201

Textbooks:

• Russell A. Gordon Real Analysis - A First Course (Second Edition), ISBN: 0-201-43727-9
• Theodore W. Gamelin, Robert Everist Greene Introduction to Topology (Second Edition), ISBN 0-486-40680-6

Instructor:

Barbara Kaskosz, Lippit Hall 202J, 874.4445, bkaskosz@math.uri.edu.
Office hours: Fridays 10 am - 1 pm.

MTH 436 is a continuation of MTH 435. We will continue studying the precise mathematical theory behind Calculus. As in MTH 435, the emphasis will be on mathematical precision and strenthening your skills in proving theorems. We will again study fundamental concepts of Calculus that you already know from the intuitive point of view. This time topics include the Riemann integral, sequences of functions, series of constants, series of functions, power series, and the Taylor series.

Many theorems in analysis that we will prove are particular cases of more general theorems in metric spaces. Therefore, we will continue our study of metric spaces.

Current Downloads

You will find here current downloads: solutions to homework problems, additional class notes etc.

Exams and Evaluation

As in MTH 435, we will have two exams during the semester, Friday March 6, 3-5 and Friday April 17, 3-5. The final exam will be scheduled by the Office of Enrollment Services based on the time your section meets. The schedule will be posted at: Final Exam Schedule. According to this grid, our exam is scheduled for Monday, May 4, 3 pm - 6 pm.

Your grade will be based upon a possible total of 500 points, as follows:

• Exam 1 -- 100 points,
• Exam 2 -- 100 points,
• Homework -- 150 points,
• Final exam -- 200 points,

Weekly homework will be assigned in class.

Topics

We will begin with few remarks on cardinality, and a few remarks and theorems in complete metric spaces. Then we will move to Real Analysis. After that we will study more topics in metric space topology.

REAL ANALYSIS

Lipschitzian functions
Relationships between Lipschitzianity, uniform continuity and differentiabilty
Differentiability of monotone functions

Partitions, refinements, the definition of the Riemann integral
Properties of the integral
Sufficient conditions for Riemann integrability
The First Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus
Sequences of functions, pointwise convergence, uniform convergence
Uniform convergence and continuity
Convergence of integrals

Series of constants, convergence
Properties of series, absolute convergence
Convergence tests
Alteranting series, Leibnitz Theorem
Limit inferior, limit superior

Series of functions
Convergence, uniform convergence
Weierstarss M-Test
Series of integrals
Power series, radius of convergence
Taylor polynomials, Taylor series

METRIC SPACES

Open covers, subcovers
Compactness
Sequential compactness
Total boudedness
Characterizations of compactness in metric spaces
Compact subspaces of Rⁿ
Heine-Borel Theorem
Separable spaces
Continuous Functions
Characterizations of continuity in metric spaces
Continuous mappings on compact spaces