18.100A: Introduction to Analysis

Instructor:

Enno Lenzmann
MIT, Department of Mathematics
Room 2-230
Email: lenzmann at math.mit.edu
Office hours: Tue, 2-3pm and Thu, 3-4pm, Room 2-230.
Grader's office hours: Wed, 2:30-3:30pm, Math Major Lounge (Room 4-174).

Time and Place:

Monday, Wednesday, and Friday, 11-12:00pm.
Room 2-131

Outline of the Course

The course 18.100A is a basic introduction to analysis, designed for students with a fair knowledge of calculus and a strong interest in its rigorous formulation. This course primarily focuses on developing mathematical proofs (encountered in one-variable calculus) and learning how to write them up. The subject matter of the first 2/3 of 18.100A roughly corresponds to Chapters 1-21 of Arthur Mattuck's monograph "Introduction to Analysis" which also serves as the course's textbook. More precisely, we will cover the following topics.

Sequences and limits.
The real numbers and their completeness property.
Infinite series and convergence criteria.
Continuity.
Differentiability of functions on the real line.
The Riemann integral.

Time permitting, the last 1/3 of 18.100A tries to convey some more advanced concepts of analysis as follows.

Uniform convergence and interchanging limits.
Basic point-set topology and compactness.

Tentative Syllabus

A detailed and tentative syllabus is given here.

Textbook

Arthur Mattuck: Introduction to Analysis. Prentice Hall, 1999. Further suggested reading will be announced in class.

Homework and Exams

There will be weekly homework assignments (except for spring break and weeks having a midterm), two midterms, and one final exam. Each counts about 1/3.

Problem Sets

Problem Set 1. Solutions to Problem Set 1.
Problem Set 2. Solutions to Problem Set 2.
Problem Set 3. Solutions to Problem Set 3.
Practice Midterm 1.
Problem Set 4. Solutions to Problem Set 4.
Problem Set 5. Solutions to Problem Set 5.
Problem Set 6. Solutions to Problem Set 6.
Problem Set 7.