Math 641 Real Analysis

Fall 2019, UAF

Instructor:
   Ed Bueler
   Chapman 306C
   office hours: bueler.github.io/OffHrs.htm
   elbueler@alaska.edu

Class:
   MWF 1--2 pm
   T 2--3 pm problem session (TENTATIVE)
   Chapman 206
   CRN: 75688

Textbook: N. L. Carothers, Real Analysis, Cambridge University Press 2000

Course Topics:

Math F641 is roughly divided into two portions, one on classical analysis from the 19th century, including metric and (some) function spaces, and one on the theory of Lebesgue integration developed in the early part of the 20th century. Specific theorems include the Banach contraction mapping (fixed-point) theorem, the existence of nowhere-differentiable continuous functions, the Weierstrass Theorem, Lebesgue’s Monotone and Dominated Convergence Theorems, and the completeness of the L_p spaces.

The UAF catalog says the course covers "General theory of Lebesgue measure and Lebesgue integration on the real line. Convergence properties of the integral. Introduction to the general theory of measures and integration. Differentiation, the product measures and an introduction to LP spaces." However, the course has evolved to be more comprehensive and somewhat less Lebesgue-oriented.

Goals and Outcomes:

The three major goals of real analysis at the graduate level:

To have a firm understanding of the underpinnings of calculus, especially as preparation for teaching calculus.
To understand integration, function spaces, and convergence as they will appear in studying differential equations, numerical analysis, and applications of mathematics.
To prepare for research in any field of mathematics which uses integration and function spaces, including functional analysis, complex analysis, numerical analysis, and probability.

This course is in the core mathematics graduate curriculum. Successful completion is preparation for the real analysis portion of the MS Comprehensive exam.

Course Website:

bueler.github.io/M641F19/

The site also has a daily schedule of topics which will be updated on an ongoing basis to reflect which topics were actually covered each day. The due dates for homework, and the dates of in-class exams, will be in this schedule.

Homework:

Weekly homework forms 40% of your score for the course. It consists mostly of proofs, but there will also be computations of examples and counter-examples, and some visualization (sketches). You are encouraged to work with other students, and seek help from the instructor, on homework problems. However, the work you turn in must be your own.

Homework will be due at the beginning of class on the announced date, and late homework will not be accepted. Homework assignments and their due dates will regularly be posted at the website (bueler.github.io/M41F19/).

On each assignment, one problem for each student will be identified as that student's "LaTeX problem". This must be sent electronically to me by the due date. I will suggest revisions immediately. Then an edited version must be sent within 24 hours. The edited version will then be included in the solution set I release. See the website for LaTeX help and materials.

Exams:

In the second week there will be a 30 minute Early Quiz so that students can see what in-class assessments will look like. There will be two in-class Midterm Exams, each a full hour, covering recent material. The in-class, two-hour Final Exam will have problems from the whole semester.

How Your Grade is Determined:

Portion
Homework
Early Quiz
Midterm Exam 1
Midterm Exam 2
Final Exam

Percentage
40%
5%
15%
15%
25%

Dates
weekly
in class Wed. 4 Sept.
in class Fri. 11 Oct.
in class Fri. 22 Nov.
in class Fri. 13 Dec., 1--3pm

A numerical score will be assigned for each homework and exam, and they will be weighted as above. Based on your raw score total, I will assign grades according to the following schedule:

90 - 100 % = A, 79 - 89 % = B, 68 - 78 % = C, 57 - 67 % = D, 0 - 56 % = F.

This schedule is a guarantee. I reserve the right to increase your grade above these ranges based on the actual difficulty of the work and/or upon average class performance.

Policies:

The Dept of Mathematics and Statistics has reasonable policies on incompletes, late withdrawals, early final examinations, etc.; see www.uaf.edu/dms/policies. You are covered by the UAF Student Code of Conduct. I will work with the Office of Disabilities Services (208 WHIT, 474-5655) to provide reasonable accommodation to students with disabilities.

Prerequisites:

An undergraduate proof-based course in real analysis, or permission of the instructor. Officially: MATH 401 Introduction to Real Analysis.