Math 305b - Real Analysis - Spring 2017 Yale

# Real Analysis

## Spring 2017

### Instructor: Guy Wolf (guy.wolf@yale.edu)

#### Peer tutor: Luke Peilen (luke.peilen@yale.edu)

The field of real analysis focuses on mathematical analysis of real-valued functions of real variables, which critically depends on a robust notion of integration. Unfortunately, classic notions developed by Riemann and Darboux are limited to (piecewise) continuous functions, and are insufficient for a general theory. In this course we will generalize these notions using Lebesgue's theory, which provides a more delicate handling of integration by using a "total length" measure of subsets of the real line. We will explore the powerful properties of Lebesgue integration, as well as its relation to Riemann integration. We will then use Lebesgue integrals to consider function spaces and introduce a rigorous mathematical notion of "almost everywhere". Finally, one of the most important applications of Lebesgue's theory is in enabling the decomposition of periodic functions into a series of sines and cosines, which is known as Fourier series. This decomposition has tremendous significance both in theory and practice (e.g., signal processing), and we will spend a significant amount of time studying some of its properties & applications.

Homework assignments will be given on a weekly or bi-weekly basis, depending on progress in class. Skipping the submission of at most two (2) assignments is allowed, although full submission is encouraged. If less than two assignments are skipped, the lowest scores will be dropped (up to two skipped/dropped ones) when computing the final grade. Further information will be given in class.

The final grade will be based on homework assignments during the semester and a final exam. The exact ratio will be announced in class, but both parts will be significant.

### Meetings

#### Lectures:

Mondays & Wednesdays 1:00-2:15, LOM 215

#### Office Hours:

Instructor: Mondays 4:00-5:00 PM, AKW 103
Peer tutor: Tuesdays, 6:00-8:00 PM, & Sundays, 1:00-3:00 PM, DL 405 (or by appointment)

### Final exam:

The final exam will take place on Saturday, May 6th, 2:00 PM at WTS A60.

Here are a few note about the exam:

1. The exam will include all the materials learned in the course. You are expected to know all the theorems, results, and proofs shown in class and in exercises.
2. You can bring with you one handwritten double-sided page of A4 or Letter size. Please use reasonable size for your handwritten notes on it, since a magnifying glass is not included in the allowed equipment for the test.
3. As you can imagine, any discovered attempt at cheating will result in 0 points on the exam.