### Math 528B, Banach and Hilbert Spaces, Spring 2018

Information TypeData
Meeting TimeMWF, 1:00—1:50
Meeting Room MATH 514
InstructorProfessor Marek Rychlik
OfficeMathematics 605
Emailrychlik@email.arizona.edu
Telephone1-520-621-6865
Homepage http://alamos.math.arizona.edu/math528
Homepage (Mirror)http://marekrychlik.com/math528

## Office Hours

PersonnelDay(s) of the WeekHourRoomComment
Marek RychlikM2:00pm—2:50pmMathematics 605 Regular Office Hours in my office
Marek RychlikW2:00pm—3:00pmMathematics 605 Regular Office Hours in my office
Marek RychlikF2:00pm—3:00pmMathematics 605 Regular Office Hours in my office

## Required Texts

Functional Analysis , Edition: Any , Author: Lax, Peter D. , ISBN: 9780471556046, e-ISBN: 9781118626740, Publisher: Wiley, Date published: 04 Apr, 2002, Pages: 509, required.

## Required Examinations And Other Grade Components

• One in-class Midterm, worth 20% of the course grade.
• One take-home Midterm, worth 20% of the course grade.
• A take-home Final Exam worth 30%.
• Written homework worth 30% (see below for details).

Homework, and take-home tests, shall be submitted as a typed paper, with the exception of these graphs and figures which cannot be easily drawn with software. Required software for preparing the homework paper is $$\LaTeX$$. The work shall be submitted electronically, as a PDF document, through D2L, using the Dropbox feature of D2L.

## Prerequisites

1. Math 528A (or equivalent coursework)
2. MATH 527B or MATH 523A (or equivalent coursework)

## Homework

Homework is assigned throughout the semester. Two types of homework will be assigned:

1. Homework which consists of selected exercises in the book.
2. Custom homework created by the instructor.

TBA.

## Overall Course Objectives and Expected Learning Outcome

1. Learn in detail the fundamental concepts of functional analysis, in particular, of Banach spaces and Hilbert spaces.
2. Learn how Banach and Hilbert spaces are used throughout many areas of mathematics, including mathematical analysis, ordinary and partial differential equations.
3. Gain and improve problem-solving abilities utilizing notions learned in the course, including norms, seminorms, convex sets, Banach and Hilbert spaces, linear operators, spectral decomposition, weak and week-* convergence, and many others.

## Course Outline

Week Dates Topics Sections Covered
1Jan 10—Jan 12 Review of weak- and weak-*-sequential convergence Riesz-Kakutani Representation Theorem (Chapter 8, Theorem 14). Chapter 10, Chapter 8
2Jan 17—Jan 19 Applications of weak convergence. Approximations of the delta function by continuous functions. Divergence of Fourier series. Approximate quadrature. 11.1, 11.2, 11.3
3Jan 22—Jan 26 Weak and strong analyticity of vector-valued functions. Existence of solutions of partial differential equations. The representation of analytic functions with positive real part. The weak and weak* topologies. 11.4, 11.5, 11.6, Chapter 12
4Jan 29—Feb 2 Locally convex topologies and the Krein-Milman Theorem. Separation of points by linear functionals. The Krein-Milman Theorem. Chapter 13, 13.1, 13.2
5Feb 5—Feb 9 The Stone-Weierstrass Theorem. Choquet's Theorem. Chapter 13, 13.3, 13.4
6Feb 12—Feb 16 Positive functionals. Convex functions. Completely monotone functions. Theorems of Caratheodory and Bochner. Chapter 14, 14.1, 14.2, 14.3, 14.4
7Feb 19—Feb 23
8Feb 26—Mar 2
9Mar 5—Mar 9 Spring Recess NO CLASSES
10Mar 12—Mar 16
11Mar 19—Mar 23 Banach algebras. Resolvent. Spectral radius. Functional calculus involving complex integration. Spectral mapping theorem. Chapter 17
12Mar 26—Mar 30 Spectral theory. Spectral measures. Projection values measures. Chapter 31
13Apr 2—Apr 6 Spectral theory. Spectral measures. Projection values measures. Classification of symmetric operators up to unitary equivalence. Chapter 31
14Apr 9—Apr 13 Spectral theory. Spectral measures. Projection values measures. Classification of symmetric operators up to unitary equivalence. Chapter 31
15Apr 16—Apr 20 Banach Algebras. Chapter 18, 19
16Apr 23—Apr 27 Banach Algebras. Chapter 18, 19
17Apr 30—May 2 Banach Algebras. Review. Chapter 18, 19
18May 7 An alternative Final Exam. Chapter 31, 18, 19.

## Course Policies

### Attendance Policy

Students are expected to attend every scheduled class and to be familiar with the University Class Attendance policy as it appears in the General Catalog. It is the student's responsibility to keep informed of any announcements, syllabus adjustments or policy changes made during scheduled classes.

### Expected Classroom Behavior

Students are expected to behave in accordance with the Student Code of Conduct and the Code of Academic Integrity. The guiding principle of academic integrity is that a student's submitted work must be the student's own. University policies can be found at http://policy.arizona.edu/academic.

### Threatening Behavior

See http://policy.web.arizona.edu/threatening-behavior-students. No prohibited behavior will be tolerated.

Students who miss the first two class meetings will be administratively dropped unless they have made other arrangements with the instructor.

### Missed Exams

Students are expected to be present for all exams. If a verifiable emergency arises which prevents you from taking an in-class exam at the regularly scheduled time, the instructor must be notified as soon as possible, and in any case, prior to the next regularly scheduled class. Make-up exams and quizzes will be administered only at the discretion of the instructor and only under extreme circumstances. If a student is allowed to make up a missed exam, (s)he must take it at a mutually arranged time. No further opportunities will be extended. Failure to contact your instructor as stated above or inability to produce sufficient evidence of a real emergency will result in a grade of zero on the exam. Other remedies, such as adjusting credit for other exams, may be considered.

### Accessibility and Accommodations

Disabled students must register with Disability Resources and be identified to the course instructor through the University's online process in order to use reasonable accommodations.

It is the University's goal that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact Disability Resources 520-621-3268 to establish reasonable accommodations.

Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable.

### Policy on the grade of "I" (incomplete)

The grade of "I" will be awarded if all of the following conditions are met:

• The student has completed all but a small portion of the required work.
• The student has scored at least 50% on the work completed.
• The student has a valid reason for not completing the course on time.
• The student agrees to make up the material in a short period of time.
• The student asks for the incomplete before grades are due, 48 hours after the final exam.

## Disclaimers

### Changes to the Syllabus

The information contained in the course syllabus, other than the grade and absence policies, is subject to change with reasonable advance notice, as deemed appropriate by the instructor.