Functional Analysis (BSc)

6 EC

Semester 2, period 4, 5

5122FUAN6Y

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

Course manual 2018/2019

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.

Objectives

At the end of this course 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.
  • has knowledge of (orthogonal) bases and (orthogonal) projections in Hilbert spaces, adjoint operators, and the Riesz representation theorem,
  • 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,
  • is familiar with the concepts of the spectrum of an operator, compact operators, and their spectral theory,
  • is able to solve elementary and more advanced exercises about all aforementioned concepts,
  • has been introduced to applications of functional analysis for solving integral and differential equations.

Teaching methods

    • Lectures
    • Exercise classes

    Learning activities

    Activiteit

    Aantal uur

    Lectures

    26

    Exercise classes

    26

    Mid-term exam

    3

    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 6.4, an other arrangement will be proposed in consultation with the study advisor.

    Assessment

    Item and weight Details

    Final grade

    3 (30%)

    Tussentoets

    2 (20%)

    Huiswerk

    5 (50%)

    Eindtentamen

    		Must be ≥ 5

    Bij het huiswerk vervallen de laagste twee cijfers.

    Inspection of assessed work

    The manner of inspection will be communicated via the lecturer's website.

    Assignments

    Exercises and other information can be found on the website: https://staff.fnwi.uva.nl/r.p.stevenson/funcanal2019.html 

    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

    Weeknummer Onderwerpen Studiestof
    1 Normed spaces   Ch. 1,2
    2 Inner Product Spaces, Hilbert Spaces  Ch. 3
    3 Linear Operators  Ch. 4
    4 Linear Operators  Ch. 4
    5 Linear Operators Ch. 4
    6 Duality and the Hahn–Banach Theorem  Ch. 5
    7 Duality and the Hahn–Banach Theorem  Ch. 5
    8 Mid-term exam  
    9 Duality and the Hahn–Banach Theorem  Ch. 5
    10 Duality and the Hahn–Banach Theorem  Ch. 5
    11 Linear Operators on Hilbert Spaces   Ch. 6
    12 Compact Operators  Ch. 7
    13 Integral and Differential Equations  Ch. 8
    14 Integral and Differential Equations  Ch. 8
    15 Zie website cursus  
    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

    • prof. dr. Rob Stevenson