*Section 2 - Spring 2008*

*Lectures:* TuTh 330-5pm, Room *2 Evans*

*Instructor:* Jan Reimann

*Office:* 705 Evans Hall

*Office hours:* We 3-4pm, Th 9-11am, 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 has 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 tab "*MATH 104 S2 Sp08*".

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.

The textbook for the course is be *Pugh, Real Mathematical Analysis* (Springer). We will follow it more or less closely. We will cover roughly Chapters 1 to 4 (we will leave out some rather advanced material in Chapters 2 and 4).

We will not have the time to go over every argument in detail. Often you will not understand and argument right away, so it is mandatory that you follow up on the lectures using your class notes and the book. Furthermore, you should prepare for the next lecture by reading ahead. I will often give reading assignments. I will also post the material for the next lecture in the course calendar on bSpace.

*Supplementary reading.* There are many good analysis books. 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.

There will be *two midterms* in class: the first on *Tuesday, Feb 19*, the second on *Tuesday, Apr 8*. Note that the first midterm is before the five-week drop-out deadline.
If you miss one of the midterms with a valid excuse, the other midterm score will count double. There will be *no makeup exams*.

The final exam is held on *Thursday, May 22, 1230-330pm*, room TBA

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

Homework will be assigned on Tuesday and will be *due on the
following Tuesday* in class. Homework will be graded and the
two lowest scores will be dropped. Late homework will not be accepted. There will be *no exception* to this rule. Of course it may happen that you cannot turn in homework because you were ill or for some other valid reason. This is why the two lowest scores will be dropped.

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 has to 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.)

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

There is a GSI for this course, *Cinna Wu*. Her office hours will be announced in the first week of classes.