## Introduction to Analysis

### Overview

Lectures: MWF 1-2pm, Room 3111 Etcheverry

Instructor: Jan Reimann

Office: 705 Evans Hall
Office hours: Mo 10:30 - 12, We 4-5:30, and by appointment

Email:
Personal Website: http://math.berkeley.edu/~reimann/

Course webpage: This site is mainly for documetary purposes. The course material (homework, sample exams, etc.) will be posted on bSpace. The course site will also have a discussion tool and a chat room. I encourage you to make use of it. Go to http://bspace.berkeley.edu and log on with your Calnet ID. Then select the accordant tab.

### Course Description

In this course we will study the foundations of real analysis. This means we will get acquainted with the real number system, how real numbers can be constructed from rational numbers via cuts, and what algebraic properties they have. At the very heart of analysis lies the ``infinitesimal'' behavior of functions. To get a handle at this, we will introduce the notion of convergence. We will see that the real numbers are superior to the rationals in the sense that they have better closure properties with respect to convergence of sequences.

A thorough understanding of sequences and convergence is a prerequisite for any higher level mathematics.

We will also see how this notions can be treated in a more general spaces, not necessarily based on real numbers, namely metric spaces. This will give us a first glimpse at how mathematics uses generalization. Namely, in order to define what it means for a sequence to converge we only need a notion of distance. This is exactly what a metric supplies. Therefore, a large part of the course will be devoted to the study of metric spaces and their topology. From experience, the step from real numbers to metric spaces is the hardest part of the course, if not of studying mathematics at all. But it is indispensable for further studies, and I think it is an advantage of the textbook that it does this step as early and as elaborate as possible.

After having mastered the topology of metric spaces, we will return to more familiar realms and develop a rigorous theory of differentiation and integration. The results we will prove should be known from calculus, the proofs will be the new part.

In the final part of the course, we will study spaces of functions. This means, that instead of studying the convergence of sequences of real numbers, we will study sequences of functions. Uniform convergence and power series will be the central notions of this section.

### Literature

The textbook for the course is be C.~C. Pugh, Real Mathematical Analysis. We will follow it more or less closely. We will cover most of the material from Chapters 1-4.

Supplementary reading. There are many good analysis books out there. There is one established classic, Rudin's Principles of Mathematical Anaylsis. This is a very good book and certainly worth to read, as it is a brilliant example of clear mathematical writing. I chose Pugh's book as the textbook because (1) it is significantly cheaper and (2) it has a more visual approach to metric spaces, which I think is quite helpful in mastering the step from real numbers to abstract metric spaces.

### Exams

There will be two midterms in class: the first on Monday, September 24, the second on Wednesday, November 7. Note that the first midterm is before the five-week drop-out deadline.

The final exam is held on Monday, December 17, 5-8 pm.

All exams will be closed book exams. No cheat sheets! Bring your blue books.

The final grade will be determined as follows: 20% homework, 20% each midterm, 40% final exam.

### Homework

Homework will be assigned on Wednesday and will be due on the following Wednesday in class. Homework will be graded and the lowest score will be dropped. Late homework will not be accepted.

A note on academic honesty: Collaboration among students to solve homework assignments is welcome. This is a good way to learn mathematics. So is the consultation of other sources such as other textbooks. However, every student should hand in an own set of solutions, and if you use other people's work or ideas you should indicate the source in your solutions.
(In any case, complete and correct homework receives full credit.)