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Functional Analysis 34 (MATH M6202)

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Administrative Information

  1. Unit number and title: MATH M6202 Functional Analysis 34
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2009/10
  6. Unit Organiser: Thomas Jordan
  7. Lecturer: Felipe Ramirez and Thomas Jordan
  8. Teaching block: 2
  9. Prerequisites: MATH 20200 Metric Spaces 2

Unit aims

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 area. It also aims to equip students with independent self-study and presentation-giving skills. 

General Description of the Unit

This unit sets out to explore some core notions in functional analysis. Functional analysis originated partly in the study of integral equations. It forms the basis of the theory of operators acting in infinite dimensional spaces. It is helpful in analysing trigonometric series and can be used to make sense of the determinant of an infinite-dimensional matrix. It has found broad applicability in diverse areas of mathematics (for example, spectral theory). Students will be introduced to the theory of Banach and Hilbert spaces. This will be followed by an exposition of four fundamental theorems relating to Banach spaces (Hahn-Banach theorem, uniform bounded-ness theorem, open mapping theorem, closed graph theorem). The unit may also include some discussion of the spectral theory of linear operators.

Relation to Other Units

This is a Level 7 version of the Level 6 unit Functional Analysis 3, and students may not take both units.  See Assessment Methods for the differences.

Teaching Methods

Lectures (30), recommended problems, guided reading and presentation.

Learning Objectives

By the end of the unit, students will

  • understand basic concepts and results in functional analysis;
  • be able to solve routine problems;
  • have developed skills in applying the techniques of the course to unseen situations;
  • have acquired independent self-study skills through guided reading;
  • have acquired presentation-giving skills.

Assessment Methods

Formative assessment will consist in a number of marked homeworks.

The final assessment mark for Functional Analysis 34 will be made up as follows:

  • 10% from a 20-minute presentation, based on independent guided reading and assessed by the unit director.
  • 90% from a 2½-hour examination in May/June consisting of FIVE questions. (It will be identical to the Level 6 Functional Analysis examination.) A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Award of Credit Points

Either passing the unit (mark of at least 50), or a mark of at least 30 together with satisfactory attempts at the homeworks and presentation.

Transferable Skills

Deductive thinking; problem-solving; mathematical exposition; presentation skills

Texts

The course will follow portions of the text Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley & Sons (1989).

The following textbooks may also be useful,

W. Rudin, Functional Analysis

N. Young, An Introduction to Hilbert Space

Syllabus

Banach spaces: bounded linear operators; bounded linear functionals; dual space

Hilbert spaces: orthogonal complement; total orthonormal set; representation of functionals on Hilbert spaces; Hilbert adjoint operator; self-adjoint, unitary and normal operators

Fundamental theorems for normed and Banach spaces: Zorn's Lemma; Hahn-Banach Theorem; Category Theorem; Uniform Boundedness Theorem; strong and weak convergence; convergence of sequences of operators; Open Mapping Theorem; Closed Graph Theorem