Ma 108a:  Classical Analysis (Fall 2016-17)

## ANNOUNCEMENTS

Graded homework and quizzes will be returned to you in recitation section.  If for some reason you do not collect your classwork in recitation section, you have the option to pick up your graded work in the Section boxes located on the 2nd floor of Sloan.

However, due to the FERPA privacy regulations, we cannot place graded work in a public location, without your permission.

Please fill out the form below to indicate how you prefer your uncollected graded classwork to be handled.  Remember that the first opportunity to collect your work is in your recitation section.

## COURSE DESCRIPTION

Many of the results encountered in calculus seem to follow common themes, even though they are usually formulated as different topics.  As an example think of the various notions of convergence typically discussed in a calculus class:  limits of a sequence of numbers, point-wise versus uniform convergence of sequences of functions, Riemann integrability etc.

The interrelation does become very clear when viewing the objects under consideration as elements of more abstract spaces equipped with some "notion of closeness" (topology).  The goal of this class is to develop some of the frameworks ("metric spaces") allowing for this more general point of view, which will naturally lead to many of the questions of classical real analysis.

The first term of Ma 108 covers the following topics in real analysis:

• Ordered sets, upper and lower bounds, least upper bound axiom, the construction of real numbers.
• Topology of metric spaces.  Open, closed, bounded and compact sets, perfect and connected sets.  Sequences in metric spaces.
• The Heine-Borel Theorem.  The Bolzano-Weierstrass Theorem.  Compact metric spaces.  Normed linear spaces.  Completeness.  Banach spaces.  Contraction Mapping Principle and applications.
• Sequences and series of functions.  Uniform convergence.  Equicontinuity.  The Arzela-Ascoli theorem.  The Weierstrass approximation theorem.
• Bounded linear transformations.  The derivative as a linear transformation.  Taylor theorem in $$\mathbb{R}_n$$.

## PREREQUISITES

Calculus of one and several variables.

## SCHEDULE

MWF, 11:00 - 11:55, 151 Sloan.

## INSTRUCTORS

Semra Demirel-Frank
HARRY BATEMAN RESEARCH INSTRUCTOR IN MATHEMATICS
174 Sloan
626-395-4354

## TA's

Connor Meehan, Pooya Vahidi Ferdowsi

## OFFICE HOURS

Semra Demirel-Frank: Monday 1-2pm
Connor Meehan: Wednesday 5pm (Sloan 153)
Pooya Vahidi Ferdowsi: Tuesday 5pm (Sloan 280)

## POLICIES

### Homework

Homework is worth 30% of the grade. Each problem set is worth the same, except your homework with the lowest score, which will be dropped.

Problem sets are available before Saturday evening on the class web page. They should be turned in on Thursday at 4:00 PM in the Math department's drop box for Math 108a. They will be available for pick-up on Wednesday in the Math department's pick-up box for Math 108a. Sample solutions will be available on the webpage after the due date.

No late work will be accepted or extensions granted. (In exchange your lowest homework score will be dropped.)

You are encouraged to discuss homework problems with one another, however the solutions must be written up in your own words.

Due to federal privacy policies, each assignment is to be submitted with a cover sheet, which carries nothing other than the student's name (first & last name).

### Exams

There are two exams, a midterm and a final each worth 35% of the grade. The respective dates and times will be announced.  The final exam is cumulative.

The midterm and final exams are timed, non-collaborative, closed-text exams, the exact policy for which will be announced at that time.

TBA

## TEXTBOOKS

Rudin, Walter, Principles of Mathematical Analysis, 3rd Edition, 1976, McGraw-Hill Science/Engineering/Math, ISBN:  007054235X.

Date Description

## HOMEWORK

Due Date Homework Solutions
6 October Set no. 1
13 October Set no. 2
20 October Set no. 3
27 October Set no. 4
10 November Set no. 5
3 November Midterm
17 November Set no. 6
28 November Set no. 7, Part 1
1 December Set no. 7, Part 2