Functional Analysis (BSc)

6 EC

Semester 2, period 4, 5

5122FUAN6Y

Owner Bachelor Wiskunde
Coordinator dr. Chris Stolk
Part of Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Bachelor Wiskunde, year 3Dubbele bachelor Wis- en Natuurkunde, year 3Bachelor Bèta-gamma, major Wiskunde, year 3Dubbele bachelor Wiskunde en Informatica, year 3

Course manual 2021/2022

Course content

Functional analysis concerns the analysis of linear spaces and mappings in infinite dimensional spaces. Other than in finite dimensional spaces the topology on these spaces plays a crucial role. So in this sense functional analysis can be viewed as a combination of linear algebra and analysis. The study of differential and integral equations has been the driving force in the development of the abstract functional analysis.

Study materials

Literature

  • Bryan P. Rynne and Martin A. Youngson, 'Linear Functional Analysis', 2nd Edition, Springer. The book can be accessed free of charge via Springerlink (see the UvA database selector).

Objectives

  • The student is familiar with the concepts of normed linear spaces, inner product spaces, and their complete variants, being the Banach and Hilbert spaces, bounded linear mappings, and normed linear spaces of those, in particular dual spaces and reflexivity.
  • The student has knowledge of (orthogonal) bases and (orthogonal) projections in Hilbert spaces, adjoint operators, and the Riesz representation theorem,
  • The student understands the main theorems in functional analysis such as the Hahn-Banach extension theorem, the open mapping theorem, the closed graph theorem, and the uniform boundedness principle,
  • The student is familiar with the concept of a compact operator
  • The student is able to solve elementary and more advanced exercises about all aforementioned concepts,

Teaching methods

  • Lecture
  • Self-study
  • Lectures
  • Exercise classes

Learning activities

Activiteit

Aantal uur

Lectures

26

Exercise classes

26

Mid-term exam

2

Final exam

 3

Self-study

110

Attendance

Programme's requirements concerning attendance (OER-B):

  • Each student is expected to actively participate in the course for which he/she is registered.
  • If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
  • It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
  • In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
  • In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article A-6.4, an other arrangement will be proposed in consultation with the study advisor.

Assessment

Item and weight Details

Final grade

0.3 (30%)

Deeltoets

0.5 (50%)

Tentamen

0.2 (20%)

Homework and in-class tests

1 (17%)

Homework 1

1 (17%)

Homework 2

1 (17%)

Homework 3

1 (17%)

Homework 4

1 (17%)

In-class test 1

1 (17%)

In-class test 2

Every two weeks there will be a homework assignment or an in-class test. With M, T being the grades for the mid-term and final exam, respectively, and W the average of the grades for the homeworks and in-class tests without the lowest one, the final grade will be given by the usual rounding of 0.2*W+0.3*M+0.5*T

In case of a retake, the final grade is given by the grade of the retake.

Inspection of assessed work

The manner of inspection will be communicated via the digitial learning environment.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Weeknumber Topics and chapters from the book
1 Ch. 1 Preliminaries, Ch. 2 Normed spaces 
2 Ch. 2, Ch. 3 Inner product spaces, Hilbert spaces
3 Ch. 3
4 Ch. 3, Ch. 4 Linear operators
5 Ch. 4
6 Ch. 4, Ch. 5 Duality and the Hahn–Banach theorem
7 Ch. 5
8 Mid-term exam
9 Ch. 5
10 Ch. 5
11 Ch. 5, Ch. 6 Linear operators on Hilbert spaces 
12 Ch. 6
13 Ch. 6, Ch. 7 Compact operators
14 Ch. 7
15 TBD. This time slot can be used if some of the weekly lectures fall on university holidays.
16 Final exam

Timetable

The schedule for this course is published on DataNose.

Honours information

There is no honours extension to this course.

Additional information

Recommended prerequisites: Linear algebra; Analysis 4; Topology; Measure Theory

Processed course evaluations

Below you will find the adjustments in the course design in response to the course evaluations.

Contact information

Coordinator

  • dr. Chris Stolk

Staff

  • Francisco Hernández Iglesias
  • Mehmet Akif Yildiz