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Homework Solutions Functional Analysis Pdf

MATH 36202/M6202 (Teaching Block 2)

Functional Analysis

  • Lecturers: Alexander Gorodnik andMin Lee
  • Class Time: Monday 2-3pm at SM3, Wednesday 11-12pm at SM4, Thursday 9-10am at SM4
  • Office hours: Monday 3-4pm and by appointment.
  • Level: H/6 and M/7
  • Credit point value: 20cp
  • Year: 15/16
  • Prerequisites: MATH 20200 Metric Spaces 2

Course Description:The unit aims to provide students with a firm grounding in the theory and techniques of Functional Analysis and to offer students ample opportunity to build on their problem-solving ability in this subject. It also aims to equip students with independent self-study and presentation-giving skills. This course sets out to explore some core notions in Functional Analysis which originated in the study of integral/differential equations and more generally equations for operators in infinite dimensional spaces. These techniques can be helpful, for instance, in analysing trigonometric series and can be used to make sense of the determinant of an infinite-dimensional matrix. Functional Analysis has found broad applicability in diverse areas of mathematics, physics, economics, and other sciences. Students will be introduced to the theory of Banach and Hilbert spaces. The highlight of the course will be an exposition of the four fundamental theorems in the Functional Analysis (Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, closed graph theorem). The unit may also include some discussion of the spectral theory of linear operators.

Lecture Notes:

Homework Problems:

Student Presentations:

Sample Exam:

  • Exam --- Solutions

    (Starting this year there is no 4/5 rule. To get full mark all problems have to be solved.)

References:

  • E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1989.
  • D. Lax, Functional Analysis, Wiley lnterscience, New York, 2002.
  • W. Rudin, Functional Analysis, McGraw-Hill, 1991.
  • N. Young, An Introduction to Hilbert Space, Cambridge University Press, 1988.
  • R. Zimmer, Essential Results in Functional Analysis, University of Chicago Press, 1990.

Meeting times: MWF, 9:00am-9:50am (MTH 0101)

Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. His office hours are M and W 10-11, or by appointment.  The course TA (for grading homework) is Matias Delgadino; his office is in room 3303 of the Math Building and you can contact him by email.

Text:Peter Lax, Functional Analysis (Wiley). The book is available at a substantial discount at amazon.com and possibly other online bookstores, so shop around.

Prerequisite: MATH 631 (real analysis).  Actually, somewhat less real analysis may do if you have a good command of basic linear algebra.

Catalog description:

Introduction to functional analysis and operator theory: normed linear spaces, basic principles of functional analysis, bounded linear operators on Hilbert spaces, spectral theory of self-adjoint operators, applications to differential and integral equations, additional topics as time permits.


Course Description:

Lax's book has a lot more in it than can be covered in one semester, but it's readable and gives you a very good feeling for what the subject is good for. I am planning to concentrate on the following topics:

  • The Hahn-Banach Theorem in all its various forms (Chapters 3-4). This subject reoccurs in another incarnation in Chapter 8.
  • Banach spaces and Hilbert spaces (Chapters 5-7). It is especially important to have a good feel for the geometry of the latter.
  • Weak compactness, a fundamental tool (Chapter 12)
  • Convex sets and the Krein-Milman Theorem (Chapters 13-14)
  • Uniform Boundedness and the Closed Graph Theorem (Chapter 15)
  • Basics of Banach Algebras, especially commutative ones (Chapters 17-19)
  • Spectral theory of self-adjoint, and especially self-adjoint compact, operators on Hilbert space (Chapters 22-23, 28-29, 31)

If you can, try to look at the "applications" chapters, even if we don't have time to cover them in class. They are some of the best parts of the book, and show you the power of the subject.

Files Tab:

Please remember to look at the Files tab in this website.  In it you will find some of the original classic papers in the subject (always instructive to read), as well as things like homework solutions.

Course Requirements and Grading Policy:

Homework will be assigned, collected, and graded, usually once a week. Homework counts for 70% of the grade. There will also be a take-home final exam due Thursday, December 18, at 10AM (the exam time for the course in the official exam schedule), counting for 30% of the grade. Grades will have the following rough meaning (modified by pluses and minuses, as appropriate):

  • A. Did most of the problems correctly, seemed to have a good idea of what's going on.
  • B. Did a fair number of problems, but had some trouble with them.
  • C. Didn't do much of the homework, or got essentially none of it right.
  • D, F. I hope these won't arise. Basically mean "hardly ever showed up, no effort at all".

It is your responsibility to turn the homework and the final exam in on time.

Attendance Policy, Academic Dishonesty, Disabilities:

I won't take attendance, but missing a lot of classes might affect your grade if you are near the borderline between two grades.  On the homework, it is OK to talk to your fellow students about the problems (in this regard you can use the Chat feature on the website), but you should write up the answers yourself.  On the take-home final exam, no collaboration is allowed, and you are expected to write out and sign the honor pledge:

I pledge on my honor that I have not given or received any unauthorized assistance on this assignment/examination. 

If you think you need accommodation for some disability, please talk to the instructor and we'll make suitable arrangements.

Course Evaluation

Please fill out the course evaluation questionnaire at https://umd.bluera.com/UMD/ before December 14th.

The syllabus page shows a table-oriented view of the course schedule, and the basics of course grading. You can add any other comments, notes, or thoughts you have about the course structure, course policies or anything else.

To add some comments, click the "Edit" link at the top.

Course Summary:

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