This course will treat the fundamentals of analysis and (linear) algebra in normed vector spaces. Its purpose is to provide students with a firm background in basic concepts which undergird modern analysis: metric and topological spaces, normed and topological vector spaces and linear algebra, differential calculus in normed linear spaces. Applications (existence and uniqueness theorems for ODE's, fixed point theorems, ...) are numerous and will figure prominently.

**Contents**

- Instructor information
- Course information
- Homework, projects and exams
- Grading Policy
- Lecture notes and assigned homework

**Instructor:**Prof. Jeremy Tyson**Office Location:**330 Illini Hall**Office Phone:**244-4132**Email:**`tyson@math.uiuc.edu`**Office Hours:**Tue 10:00-11:00, WedFri 2:00-3:00, or by appointment

**Lecture Times:**MonWedFri 11:00-11:50**Lecture Location:**145 Altgeld Hall**Course web site:**https://math.uiuc.edu/~tyson/595f05.html.

**Homework and projects:**There will be regular homework assignments, due each Monday. In addition, there will be three extended assignments/projects, one for each of the three parts of the course. Each project will be due approximately 1-2 weeks after we complete the corresponding part of the course. You are encouraged to work on the homework and projects in groups, although each student should submit a separate set of solutions. (Please indicate which other students you collaborated with, if any.)

**Exams:**There will be two one-hour midterm exams (in class) and a cumulative final exam. Some of these exams may be takehome exams.

**Grading Policy:**Grades will be computed according to the following percentages:Homework 25% Projects 10% First Midterm 20% Second Midterm 20% Final 25%

**Lecture notes:**1. Linear algebra:

- 1.1 Vector spaces, linear independence, bases, dimension;
- 1.2 Linear transformations, kernel, range, matrix representations;
- 1.3 Geometry of linear transformations: determinant, trace;
- 1.4 Spectral theory and Jordan canonical form: eigenvalues, eigenvectors, Spectral Theorem;
- 1.5 Matrix exponentiation.

- 2.1 Metric spaces;
- 2.2 Continuous functions: Lipschitz, Holder, uniform continuity;
- 2.3 Complete metric spaces and completions;
- 2.4 Function spaces and isometric embeddings;
- 2.5 A famous example: completion of C([0,1]) in the L^1 norm;
- 2.6 Compactness;
- 2.7 Fixed point theorems and applications to ODE's;
- 2.8 Introduction to topological spaces.

3. Analysis on normed vector spaces:

- 3.1 Topological and normed vector spaces;
- 3.2 Continuous linear maps and the dual space;
- 3.3 Geometry of the dual space;
- 3.4 The Hahn-Banach theorem and applications;
- 3.5 Differentiation in Banach spaces;

**Homework assignments:**Homework #1 Solutions to selected problems

Homework #3 Solutions to selected problems

Homework #5 Solutions to selected problems

Homework #7 Solutions to selected problems

- A. Non-commutative matrix exponentiation and the Baker-Campbell-Hausdorff formula,
- B. Free vector spaces and tensor products

- The Hausdorff metric on compact hyperspace