MA4MSF: Metric Spaces and Functional Analysis

Module code: MA4MSF

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 4 (Undergraduate Masters)

When you'll be taught: Semester 2

Module convenor: Dr Chris Daw, email: chris.daw@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST TAKE MA2RAT OR TAKE MA2RA2 (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA3MSF OR TAKE MA3MS (Compulsory)

Placement information: NA

Academic year: 2024/5

Available to visiting students: Yes

Talis reading list: Yes

Last updated: 21 May 2024

Overview

Module aims and purpose

The module studies analysis from a more general perspective, based on the concepts of distance. Normed, and metric spaces are introduced, and the concepts of convergence, continuity, compactness and completeness are developed in this general framework. Some exemplary applications are given. This module puts the material studied in previous courses in analysis in a simple and elegant yet general framework and provides a foundation for further courses in analysis and other areas of mathematics. The types of linear spaces we study are Banach spaces and Hilbert spaces, which are the two most fundamental structures with great importance in other parts of mathematical analysis and its applications.

The aim is to introduce students to the concepts of basic topology and functional analysis and enable them to use these in the study of appropriate problems arising in applications. To introduce the basic theory of Hilbert spaces, discuss their geometry, which generalizes the notion of the finite-dimensional Euclidean spaces, and demonstrate the effect of the dimension of the space in its study.

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Give a rigorous account of the main definitions in metric spaces; apply the notions of convergence, continuity, compactness and completeness to solve problems; give a rigorous account of the geometry of Hilbert spaces
2. Give a rigorous account of the difference between Hilbert and Banach spaces and of the role of the inner product; explain the proofs of the main theorems and how they can be applied to solve problems in other areas of mathematics

Module content

Metric spaces and normed spaces: definitions, the metric induced by a norm, examples, bounded sets, convergence of sequences, continuity of functions, sequential characterization of continuity, equivalent metrics and equivalent norms, subspaces

Completeness of metric spaces: definition, basic properties, the notion of a Banach space, proof of completeness of some important examples of metric spaces.

Compactness: sequentially compact sets, totally bounded sets in metric spaces, equivalence of sequential compactness to completeness plus total boundedness. Finite-dimensional normed spaces: equivalence of any two norms, completeness, compactness of closed bounded sets, the Riesz Lemma, characterization of finite-dimensionality by means of the compactness of the unit ball.

Hilbert spaces, examples of Hilbert spaces, orthogonality, the Riesz representation theorem, orthonormal sets, isomorphic Hilbert spaces. Banach spaces, examples of Banach spaces, bounded linear operators and functionals, the Hahn-Banach theorem, the dual space, the open mapping theorem, the closed graph theorem, the principle of uniform boundedness. Applications.

Structure

Teaching and learning methods

Lectures supported by tutorials. Learning materials (lecture notes/reading lists, tutorial problem sheets, assessments) made available via Blackboard.

Study hours

At least 50 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 50% to pass this module

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: https://www.reading.ac.uk/cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Problem sheets. 

Reassessment

Additional costs