MATH 806 - Fall 2015

September 22,  2015

Official course description. Fundamental results for normed linear spaces with special emphasis on the theory of Hilbert space, spectral theorems and application to integral and differential equations.

My view. We will explore the construction and properties of the most basic classes of functions spaces (normed spaces, Banach spaces, Hilbert spaces) and the operators defined between them. Most of the course will be focused on linear functional analysis. We will spend an important amount of time understanding the big theorems on Banach spaces while you (the students) explore the rudiments of Hilbert space theory. Then we will have a good look at the solvability theory and the spectral theory for compact operators in Hilbert space. Time permitting, we will try to fly high and have a look at algebras of operators.

Basic data.
  • Lectures: MW 5:00 PM to 6:15 PM at Ewing 203
  • Instructor: Francisco-Javier Sayas (www.math.udel.edu/~fjsayas)
  • Textbook: Alberto Bressan. Lecture Notes on Functional Analysis. AMS Graduate Studies in Mathematics, Volume 143.
  • Office hours: By appointment (see next item).
  • Problem solving sessions: Mondays 1pm-2pm in EWG336. Fridays 11am-noon in EWG336. These are to be understood as group office hours, or (equivalently) as tutored working time.
  • Website: www.math.udel.edu/~fjsayas/teaching/math806

SYLLABUS

WHAT WE WILL DO IN THIS CLASS

 
Class and study time will be divided in different activities:
  • Lecturing. At the time of each lecture you will be expected to have read a section of the book. The lectures will be self-contained but will only contain sketches of proofs or difficult details that you might find hard to understand when reading the book.
  • Quizzing. Every now and then there will be an unannounced quiz in class. Quizzes will assess on basic concepts (definitions, statements of important theorems) and mathematical techniques (one line proofs)
  • Problem solving. Homework will not be collected or graded. We will articulate ways of going through the most challenging problems so that you are prepared for tests.
  • Reading. In addition to take some time to read the sections of the book that we will be covering, I will prepare some documents where you will be challenged to read (and fill the gaps) on theory that we will see later on.
  • Exams. There will be two in-class exams and two environment-controlled take-home exams. 80% of your grade will be built (evenly) from these four tests. The remaining 20% will be derived from performance in quizzes.

Read the syllabus for my views and expectations with respect to collaboration and online search of solutions.

THE SCHEDULE


Last update. November 9, 2015

Week
 Day
 Topic (and section in the book)
 Recommended problems
1
 09/02
 Metrics, norms, and inner products (2.1)
 [2.1],[2.2],[2.3],[2.4],[2.5]
2
 09/09
 Young, Hölder, and Minkowski. L^p spaces
 [2.11],[2.14],[2.15],[2.17],[2.19]
3
 09/14
 Finite dimensional spaces (2.3).
The completion of a normed space 		[2.10],[2.17],[2.25],[2.30],  [2.37]
09/16
 Linear operators (2.2)
4
 09/21
 The Hahn-Banach Theorem (2.5) [2.20],[2.27],[2.29],[2.32],[2.33]
09/23
 Extension and separation theorems (2.5, 2.6)
5
 09/28
 The dual space and weak convergence (2.7) [2.21],[2.23],[2.24],[2.34]
09/30
 Construction of Frechet spaces (2.4)
6
 10/05
 The Baire Category Theorem (Appendix)
10/07
 The Banach-Steinhaus Theorem (4.1)
 [4.3], [4.4], [4.18]
7
 10/12
 [This class was cancelled]
10/14
 Open map and closed graph (4.2-4.3)
8
 10/19
 FIRST MIDTERM EXAM
10/22
 The adjoint of a bounded operator (4.4)
9
 10/26
 Compact operators (4.5)
 [4.1],[4.5],[4.7],[4.9],[4.12],[4.15],[4.16],[4.23]
10/28
 Spectral theory of compact operators
10
 11/02
 Orthogonal decompositions in Hilbert spaces (5.2)
 [5.2],[5.3],[5.4],[5.5],[5.6],[5.7],[5.8],[5.9], [5.10], [5.13], [5.15]
11/04
 The dual of a Hilbert space (5.3)
11
 11/09
 The Lax-Milgram lemma (5.6)
 [5.16],[5.19],[5.21],[5.23],[5.24],[5.25],[5.27],[5.28]
11/11
 Fredholm theorems (6.1)
12
 11/16
 Fredholm theorems II (6.1)
11/18
 The Hilbert-Schmidt theorem (6.2-6.3)
13
 Thanksgiving Week
14
 11/30
 The Hilbert-Schmidt theorem II (6.3)
12/02
 SECOND MIDTERM EXAM
15
 12/07

12/09