Syllabus for MATH 615, SECTION 700, Spring 2019

Introduction to Classical Analysis

Textbook: Real Analysis, by Neal Carothers, Cambridge University Press (2000).

Prerequisites: Students should have had a course in advanced calculus or real analysis.   

Course Goals:  We will cover many of the core topics in classical analysis, including the five C's: continuity, connectedness, completeness, compactness, and category. We will work through the first third of Carothers' text, Chapters 1 through 9.

Lectures: The lectures for this course will all be on-line, available on the course web site. The material for each week has been divided up into about ten lectures, which I've tried to keep under about 15 minutes each. (I haven't always succeeded.)

Homework Assignments: A homework assignment will be made in eCampus on most Fridays, due the following Friday. (No homework will be assigned for the week of spring break.) The final homework assignment will be due Friday, May 3. Homework assignments will typically consist of eight problems, worth ten points each. Work will be accepted up to a week late, though five points will be deducted for each day the assignment is late.  

eCampus: Homework will be submitted and returned through the eCampus system, and this is also where homework solutions will be posted. In addition, we'll use the eCampus system for a class discussion board. Students can log into the eCampus system using the link http://ecampus.tamu.edu, and can find help with using the eCampus system at the link http://ecampus.tamu.edu/student-help.php.  

Grades: Final grades will be determined entirely from an average of 14 homework scores. Grade ranges will be standard: 89.50-100, A; 79.50-89.49, B; 69.50-79.49, C, 59.50-69.49, D; below 59.50, F.

Learning outcomes: Students will be able to: apply the Least Upper Bound Axiom in developing fundamental properties of the real line; compute the limit superior and the limit inferior of a set of real numbers; determine the cardinality of certain infinite sets; use the Cantor set as an example and counterexample in the study of general sets; identify the properties of a metric, and rigorously establish properties of metric spaces; identify open and closed sets; identify the interior of a set; identify the closure of a set; define continuity in general metric spaces; define homeomorphisms; define the notion of connectedness, and be able to identify disconnections; identify totally bounded sets; state and apply the Contraction Mapping Theorem; understand the relationship among several different characterizations of compact metric spaces; recognize the difference between continuous and uniformly continuous functions; apply the Baire Category Theorem to establish properties of the real numbers as well as general complete metric spaces.

Make-up policy: Make-ups for exams will only be given if the student can provide a documented University-approved excuse (see University Regulations). According to University Student Rules students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise the student forfeits his or her right to a make-up.

Scholastic Dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. "An Aggie does not lie, cheat, or steal or tolerate those who do." Please refer to the Honor Council Rules and Procedures, available at the Office of the Aggie Honor System.

Copyright policy: All printed materials disseminated in class or on the web are protected by copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Students with Disabilities: The following statement was provided by the Department of Disability Services: The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, in Cain Hall, Room B118, or call 845-1637. For additional information visit http://disability.tamu.edu.

Class Schedule: We will cover the following material on the following schedule. Additional details are available on the course web site (see full listing of lectures).

Week of Monday	Material Covered
January 14	Least Upper Bound Axiom; Monotone Convergence Theorem; P-adic expansions.
January 21	Limits superior and inferior; Bolzano-Weierstrass Theorem; cauchy sequences; continuity.
January 28	Set equivalence; countable and uncountable sets; Cantor's Theorem.
February 4	The Cantor set; the Cantor function; properties of monotonic functions.
February 11	Metric spaces; normed vector spaces; limits in metric spaces.
February 18	Open sets and closed sets; interior and closure; the relative metric.
February 25	Continuity in metric spaces; homeomorphisms; the space of continuous functions.
March 4	Connectedness; disconnections; the Intermediate Value Theorem; space-filling curves.
March 11	Spring Break.
March 18	Totally bounded sets; complete metric spaces; characterizations of complete metric spaces.
March 25	Banach spaces; applications of the Contraction Mapping Theorem.
April 1	Completions; uniqueness of completions; completions of normed vector spaces.
April 8	Compactness; characterizations of compactness; uniform continuity.
April 15	Equivalent metrics; the space of bounded linear operators.
April 22	The oscillation of a function; category; The Baire Category Theorem.
April 29	No lectures. Tuesday, April 30 is the last day of campus classes. Assignment 14 is due Friday, May. 3.