Functional Analysis Kreyszig Homework

TMA4230 Functional analysis, Spring 2011




We meet first time Thursday 13 January and continue according to the following schedule.


  • Thursday 12:15 - 14:00 in F6.

  • Friday 8:15 - 10:00 in F2.

Problem session

  • Tuesday 14:15 - 15:00 in F4.

The last lecture will take place Friday 15 April.


This course will be taught in English. The course text and supplementary material are also written in English. The exam will be in English or Norwegian at the choice of the student.


The examination

The exam is oral and takes place Wednesday June 1 in room 734 and 822 in Sentralbygg 2. One hour, which includes time for the examiners to discuss the grade etc., will be scheduled for each student, so you should expect that your examination last for a maximum of 45 minutes. During these 45 minutes you will be asked questions from the syllabus (see below). You are not expected to remember every little details of every proof, but you should be able to state the main theorems and important definitions, give examples of applications of the main theorems and other important concepts, and tell something about the proofs of the main theorems. The schedule is as follows

08:00 Sigurd Storve, Christian G Frugone734
10:00 Gizat Derebe Amare, Espen Sande822
13:00 Anders Samuelsen Nordli, Karl Kristian Brustad822
15:00 Axel Byberg Fosse, Erik Korsnes, Kalliopi Paolina Koutsaki822

You should be at the assigned room no later than at the time assigned to you. As you can see, two or more students are assigned for each time. This is to take in account that some might not show up.


You are expected to know and understand the contents of Section 4.1-4.9 and Section 4.12-4.13, Chapter 7 and Chapter 9 of Introductory functional analysis with applications by Erwin Kreyszig in addition to pp. 32-43 (excluding the section “Normal spaces and the existence of real continuous functions”), 52-54 (only the section “The Banach-Alaoglu theorem”) and 61-65 (excluding the section “Holomorphic functional calculus”) of Harald Hance-Olsen's notes Assorted notes on functional analysis and the note about the Stone-Weierstrass Theorem I handed out in class (if you do not have this note, send me an email and I will then send you a pdf-file with the note, or see me in class or at my office if you prefer a paper version).

Contents of the course

Semester plan

WeekDatesSubjectsReferencesExercisesWeeklySolutions to exercises
2Jan 10-14Introduction/Review of TMA4145Chapter 1-3Noneweek1.pdf
3Jan 17-21Zorn's Lemma, Hahn-Banach theoremsSection 4.1-, 2.8.9, 2.10.8, 2.10.13, 3.10.3, 3.10.4 week2.pdfsolution2.pdf
4Jan 24-28Bounded linear functionals on C[a,b], Riesz's representations theorem, Hilbert-adjoint operatorsSection 4.4 + 3.8-3.9 4.1.2, 4.2.3, 4.2.5, 4.2.6, 4.2.10 and 2.8.12+4.3.14week3.pdfsolution3.pdf
5Jan 31 - Feb 4Adjoint operators, reflexives spaces, Baire's category theorem, uniform boundedness theoremSection 4.5-, 3.8,6, 3.8.8, 3.9.3, 3.9.10, 3.10.4 plus this exercise.week4.pdfsolution4.pdf
6Feb 7-11Strong and weak convergence, convergence of sequences of operators and functionals, open mapping theoremSection 4.8-4.9 +, 4.5.9, 4.5.10, 4.6.4 and 4.6.7 plus two extra exercises which can be found here week3.pdf.week5.pdfsolution5.pdf
7Feb 14-18Closed linear operators, closed graph theorem, topologySection 4.13 + page 32-39 of the notes 4.7.6, 4.8.1, 4.9.3 and 4.9.6 plus one extra exercise which can be found here week5.pdf.week6.pdfsolution6.pdf
8Feb 21-25Compactness, Tychonoff ’s theorem, Banach-Alaoglu theoremPage 39-43 + 52-54 of the notes4.12.5, 4.12.6, 4.12.8, 4.12.9 4.12.10, 4.13.11 and 4.13.14week7.pdfsolution7.pdf
9Feb 28 - Mar 4Stone-Weierstrass theorem, an application of Banach-Alaoglu's theorem to PDEs, spectral theory in finite dimensions and basic concepts of spectral theoryNotes on the Stone-Weierstrass theorem, note on an application of Banach-Alaoglu's theorem to PDEs, section 7.1, 7.24 exercises that can be found here week7.pdf.week8.pdfsolution8.pdf
10Mar 7-11Spectral theory for Banach algebrasSection 7.3-7.7 + page 61–65 of the notes 7.1.10 ,7.1.15, 7.2.3 and 7.2.6 plus one extra exercise which can be found here week8.pdf.week9.pdfsolution9.pdf
11Mar 14-18Spectral theory for Banach algebras, spectral properties of bounded self-adjoint linear operatorsSection 7.3-7.7 + page 61–65 of the notes, Section 9.1-9.2 7.3.4-6, 7.4.4, 7.5.1, 7.7.4 and 7.7.5week10.pdfsolution10.pdf
12Mar 21-25Spectral properties of bounded self-adjoint linear operators, positive operatorsSection 9.2-, 7.4.8, 7.4.9, 7.5.5, 7.5.9, 7.6.3 and 7.7.7.week11.pdfsolution11.pdf
13Mar 28 - Apr 1Square roots of a positive operators, projection operators, spectral families, spectral family of a bounded self-adjoint linear operatorSection 9.4-9.8 9.1.6, 9.2.9, 9.3.2, 9.3.9+10 and 9.3.11week12.pdfsolution12.pdf
14Apr 4-8Introduction to C*-algebras and graph C*-algebras, guest lecture by Takeshi Katsura about semiprojectivity and graph C*-algebras Slides9.4.8+9, 9.5.1, 9.5.3, 9.6.10+12+13 week13.pdfsolution13.pdf
15Apr 11-15Spectral family of a bounded self-adjoint linear operator, spectral representation of a bounded self-adjoint linear operator, extension of the spectral theorem to continuous functions, properties of the spectral family of a bounded self-adjoint linear operatorSection 9.8- plus two extra exercises which can be found here week13.pdf.week14.pdfsolution14.pdf

Unless mentioned otherwise the references above are to Kreyszig's book. "The notes" are Harald Hance-Olsen's notes Assorted notes on functional analysis.This plan is tentative and can (and probably will) be changed during the semester.


If you have any questions concerning the course, you are welcome to send me an email or stop by my office. You can find my contact information here.

Instructor: Demetrio Labate

When and Where

  • MEETING TIME:   MW 1-2:30,
  • OFFICE HOURS:   Mon: 12-1; Wed: 2:30-3:30 (or by appointment)

Course Description:

This course is the first part of a two-semester sequence about elementary functional analysis. The core of the course in an introduction to the theory of Hilbert and Banach spaces and the main properties of Linear Operators acting on these spaces. A number of applications from numerical analysis, harmonic analysis and PDEs and will also be presented in class. The second semester will be a more technical development of the theory of linear operators on Hilbert spaces; unbounded operators; topics from Fourier Analysis. The selection of topics for the second semester will be based, in part, on the interest and feedback from interested students.


Introductory Functional Analysis with Applications, by Kreyszig, Wiley, 1989.


Real Analysis (MA 4331 or, better, MA 6320-6321) and Linear Algebra (MA 4377). The course and the textbook do not require a specific knowledge of measure theory, so that students don't need be too concerned if they lack that background. However, a solid background on elementary linear algebra (e.g., matrices, linear independence), analysis (e.g., convergence) and topology (e.g., open/closed sets) are necessary to successfully atten this class.

Course outline:

  • Metric Spaces
  • Normed and Banach Spaces
  • Inner product spaces and Hilbert Spaces.
  • Fundamental theorems for normed and Banach spaces
  • Banach Fixed Point Theorem and its applications
  • Applications from approximation theory and harmonic analysis


  • Homework 1 - Due: 9/1
  • Homework 2: Sec.1.3, Ex 10,12,14; Sec.1.4, Ex 8; Sec. 1.5, Ex 10,13,14 - Due 9/13
  • Review Problems (for Test 1 on 9/22 - Not to be collected): Sec 1.2, Ex 7,8,9,10; Sec 1.3, Ex 12; Sec 1.4, Ex 1,2,3; Sec 1.5, Ex 6,7,8; Sec 1.6, Ex 13,14; Sec. 2.2 Ex 11,12, Sec. 2.3, Ex 10
  • Homework 3: Sec. 5.1: 10,12; Sec.2.4:1;Sec.2.5:10; also: prove that C^1[0,1] is a complete metric space - Due: 10/11 - Solution
  • Homework 4: Sec.2.6, Ex 14; Sec.2.7, Ex 5,6; Sec. 2.8, Ex 6; Sec. 2.10, Ex. 8 - Due 10/25
  • Homework 5: Sec.3.1, Ex 6; Sec.3.2, Ex 10; Sec. 3.3, Ex 8; Sec. 3.5, Ex. 8; Sec 3.6, Ex 4,10 - Due 11/8
  • Review problems for Test 2: 2.2: 11-14; 2.3:12-13; 2.6: 7,11-13; 2.7:7-10; 2.8: 9-10; 2.10: 6-7; 3.2: 7-10; 3.3: 2,6,8; 3.4:5-7; 3.5: 5-6; 3.6: 4-5; 3.8: 2,5; 3.9: 3-5; 3.10: 4-5, 8,9,11,12.
  • Review Problems for Final Exam: Sec 4.2: 2,5; Sec 4.3: 4,6,8,11; Sec 4.5: 1; Sec 4.6: 3; Sec 4.7: 8-12

Tests and Exam Dates:

The dates for the midterm exams are Wed Sept 22 and Mon Nov 15. The final exam will be posted on Thu Dec 9 and collected on Mon Dec 13, 9:30AM.


Grades will be based on homework assignments counting 40% towards the final grade, on two midterm exams counting 30% towards the final grade and one final counting 30% towards the final grade.

The grade will be determined according to a set point scale: 90%-100%: A, 80%-89%: B, 70%-79%: C, 60-69% D; F is less than 60% (+ and - will also be used).

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