# Functional Analysis Homework Solution

ABOUT THE COURSE: The catalog description for this class is: "An introduction to the standard topics of functional analysis are given. Properties of normed linear spaces, Banach spaces, and Hilbert spaces are studied. The Hahn-Banach Theorem is addressed. Spectral theory and compact operators are introduced. A knowledge of measure theory is not assumed and any needed measure theoretic results are presented in the course." As the last sentence suggests, a functional analysis class normally has a prerequisite of a graduate level real analysis sequence (ETSU's Real Analysis 1 and 2, MATH 5210-5220). So a full-blown functional analysis sequence appropriately belongs in a Ph.D. program. This is why our class is titled "Fundamentals of Functional Analysis."

## Functional Analysis Homework Solution

A NOTE ABOUT HOMEWORK: YOU MUST SHOW ALL DETAILS ON THE HOMEWORK PROBLEMS!!! Justify every step and claim you make - this is how you convince me that you know what you are doing. You may find some answers online, but these rarely sufficiently justify all steps and are unacceptable as homework solutions.

ACADEMIC MISCONDUCT: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Some of the homework problems are fairly standard for this class, and you may find proofs online or in an online version of the solutions manual. The online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit. If you copy homework solutions from an online source, then you will get no credit. These are examples of plagiarism and I will have to act on this as spelled out on ETSU's "Academic Integrity @ ETSU" webpage (last accessed 2/14/2021). To avoid this, do not copy homework and turn it in as your own!!! Even if you collaborate with someone, if you write the homework problems out in such a way that you understand all of the little steps and details, then it will be unique and your own work. If your homework is identical to one of your classmates, with the exception of using different symbols/variables and changing "hence" to "therefore," then we have a problem! If you copy a solution from a solution manual or from a website, then we have a problem! I will not hesitate to charge you with academic misconduct under these conditions. When such a charge is lodged, the dean of the School of Graduate Studies is contacted. Repeated or flagrant academic misconduct violations can lead to suspension and/or expulsion from the university (the final decision is made by the School of Graduate studies and the graduate dean, Dr. McGee).

SUPPLEMENTAL SOURCES Michael Reed and Barry Simon, Methods of Modern Mathematical Physics: I. Functional Analysis (Revised and Enlarged Edition), Academic Press (1980). This is a standard graduate level text on functional analysis (in fact, it was used at Auburn University in the late 1980s). The Sherrod Library has a copy of the 1972 version - it's call number is QC20.R37 1972. It is referred to in the class notes as simply "Reed and Simon's Functional Analysis."Online notes based on this source will be posted at Functional Analysis notes. Lokenath Debnath and Piotr Mikusinski, Introduction to Hilbert Spaces with Applications 3rd Edition, Elsevier Academic Press (2005). This book includes much of the material we cover on Hilbert spaces and operators on Hilbert spaces. It also includes applications to integral and differential equations, partial differential equations, quantum mechanics, and wavelets. Online notes based on this source are available at Hilbert Spaces and Applications Class Notes (these notes were used in teaching Applied Mathematics 1 [MATH 5610] in fall 1998). H. L. Royden and P.M. Fitzpatrick, Real Analysis, 4th Edition, Prentice Hall (2010). This is the book we use in graduate level Real Analysis 1 and 2 (MATH 5210/5220). The middle part of the book contains several topics from functional analysis (Banach spaces, Hilbert spaces, funtionals, and operators). Online notes for this part of the book are online at: Royden and Fitzpatrick notes on functional analysis topics. Eduard Prugovecki, Quantum Mechanics in Hilbert Space, 2nd Edition, Academic Press (1981). The first three chapters of this book cover Hilbert space theory, measure theory and Hilbert spaces of functions, and the theory of linear operators in Hilbert spaces. Online notes based on this source are partially available at Hilbert Spaces and Quantum Mechanics Class Notes.

Problem sets will be posted here weekly, on Wednesdays, and will be collected the following Wednesday at the beginning of class. You are encouraged to discuss the homework with your fellow students, and to collaborate on problems, but your final write-up must be your own. Please make sure that your solutions are written clearly and legibly. (Typing up solutions in LaTeX is encouraged, and is valuable practice for mathematical writing later in your career.)

An introductory graduate level course including the theory of integration in abstract and Euclidean spaces, and an introduction to the basic ideas of functional analysis. Math 5051-5052 form the basis for the Ph.D. qualifying exam in analysis. Prerequisites: Math 4111, 4171, and 418, or permission of the instructor.

The homework will be due on gradescopeon the designated day, usually Fridays. You areallowed to discuss the homework with others in the class,but you must write up yourhomework solution by yourself. Thus, you should understand the solution,and be able to reproduce it yourself. This ensures that, apart fromsatisfying a requirement for this class, you can solve the similarproblems that are likely to arise on the exams.

Course syllabus outlining the topics we'll cover, assignments, and some related references.

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Notes written by Professor Perry on multivariable differential calculus, includingthe inverse and implicit function theorems.

Course Material for Spring 2020:MA 671 Complex Analysis Course Syllabus for MA 671, Spring 2020. Class meets MWF 2:00-2:50 in CB 341. This is a basic course in complex analysis covering analytic and meromorphic functions, including Cauchy's theorems, Taylor and Laurent expansions, Residue Theorem, and conformal mappings. If time permits, we cover other topics such as analytic continuation and the principle of the argument.NEW: Hour exam 1 on Wednesday, 26 February, in class. NEW: Lecture discussions will by made via zoom starting 23 March. I'll send out the invitation about 1:30. the lecture notes will be sent to you. I'll also email homework assignments.

Class meets MWF 2:00-2:50 in CB 347.This course focuses on the theory of linear operators in Hilbert spaces. recommended text: J. B. Conway, A course in functional analysis, second edition, Springer, 1990.

Problem set 1 - you do not have to do number 5 - we'll do it on a later PS. Due in class, Wednesday, 7 September 2016.

Problem set 2. Due in class, Wednesday, 21 September 2016.

Problem set 3. Due in class, Monday, 3 October 2016.

Problem set 4. Due in class, Monday, 24 October 2016.

Problem set 5. Due in class, Wednesday, 9 November 2016.

Course Material for Spring 2016:MA507 and PHY507 Mathematical Methods of PhysicsCourse Syllabus for MA and PHY 507 001, Spring 2016.

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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.

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.

We present results in the form of student feedback from a course on functional analysis for third- and fourth-year students. Moreover, we analyze marking results from two courses on real analysis. Here, we compare tasks marked by the teacher and tasks marked by the students.

Our second experience with self-assessment was the following. During a 14-week course on real analysis, taught in 2013 for first-year students, we gave the following two exercises. Both were given as additional exercises and were credited with 20 marks. The total of regular marks was 480. The self-assessment homework thus counted as approximately 4% extra credit.

The third experiment on self-assessment was part of a 12-week course on real analysis for first-year students taught in 2018. We mention that we had a very small group of only seven students and thus an atmosphere in which the students know each other well and talk much about math, homework, exams, etc. The assessment consisted of a final exam and one longer homework assignment in the middle of the course. Both components contributed 50% to the final grade. The homework assignment consisted of 10 questions. It covered elementary logic, sets, mappings, and mathematical induction. One of the 10 questions was the following.