Math 510 Syllabus

# Math 510: Introduction to Analysis I

## Instructor: Cristina Pereyra

### M ATH 510 - INTRODUCTION TO ANALYSIS

Fall 2005
Instructor: Cristina Pereyra
E-mail: crisp AT math . unm . edu
Office: Humanities 459, 277-4147
Schedule: MWF 9-9:50am at HUM 428
Office Hours: M 2-3:30pm, W:1-2:30pm, or by appointment

Course content: This is a first graduate course on Analysis. It is also the course that prepares graduate students for the Real Analysis Qualifying. Students should have at this point enough computational background (calculus of one variable and multivariable calculus -at least 2 and 3 variables), and I expect most of you to have been exposed at least once to rigorous epsilon-delta proofs, if not you might consider taking Math 401/501 instead. I expect familiarity with the real an complex numbers.
We will start the course with a quick overview of the real and complex numbers numbers. We will then review some basic point set topology, metric spaces, compact and connected sets. Next topic are sequences and series, mostly of complex numbers, however the basic concepts of convergence will be described in the more general setting of metric spaces. We then consider functions defined and with values on arbitrary metric spaces, and we define limits, continuity, and connections to compactness and connectedness. We then specialize to real valued functions defined on an interval or the real line. We study differentiation properties and Riemman-Stieltjes integrals of such functions. At this point we are ready to study sequences and series of real-valued functions, and their interplay with integrations and differentiation, here the concept of {\em uniform convergence} is crucial. Time permitting we will study power series, Fourier series, and the classical Stone-Weierstrass approximation theorem.

Prerequisites: Advanced calculus and linear algebra, or permission from the instructor.

Required Textbook: Principles of Mathematical Analysis by Walter Rudin. MacGraw Hill Inc. Third Edition.
We will cover chapters 1-7, and time permiting parts of chapter 8.
There are many books on analysis, some are classical, some present fresher views of the subject. Reading from more than one source will enhance your learning, and will help you build the big picture.

Exams: There will be two midterms and a final exam.

Homework: Homework problems will be handed out weekly or bi-weekly, and they will be graded and returned to you promptly. Please no late homework! Problems from past real Analysis Qualifying exams will be weaved into the homework, hopefully by the end of the course you will have built a folder with solutions to most of those problems for future reference.

Grades: The final grade will be determined by homework (25%), two midterms( 50%) and a final exam (25%).

Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.

Last updated: 17 January 2005