This popular maths talk gives an introduction to various different kinds of infinity, both countable and uncountable. These concepts are illustrated in a somewhat informal way using the notion of Hilbert's infinite hotel. In this talk, the hotel manager tries to fit various infinite collections of guests into the hotel. The students should learn that many apparently different types of infinity are really the same size. However, there are genuinely "more" real numbers than there are positive integers, as is shown in the more challenging final section, using Cantor's diagonalization argument.

During the academic year 2011-12, Dr Joel Feinstein gave five optional example classes to his second-year Mathematical Analysis students on Definitions, Proofs and Examples. Dr Feinstein recorded videos of these classes (presented here) to go along with his previous videos on 'How and why we do mathematical proofs'.
These sessions are intended to reinforce material from lectures, while also providing more opportunities for students to hone their skills in a number of areas, including the following:

•working with formal definitions

•making deductions from information given

•writing relatively routine proofs

•investigating the properties of examples

•thinking up examples with specified combinations of properties

As taught in 2006-2007 and 2007-2008.

Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions.

This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include:

– norm topology and topological isomorphism;
– boundedness of operators;
– compactness and finite dimensionality;
– extension of functionals;
– weak*-compactness;
– sequence spaces and duality;
– basic properties of Banach algebras.

Suitable for: Undergraduate students Level Four

Dr Joel F. Feinstein
School of Mathematical Sciences

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.

Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

This is a module framework. It can be viewed online or downloaded as a zip file.

As taught Autumn semester 2010.

Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions.

This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include:

– norm topology and topological isomorphism;
– boundedness of operators;
– compactness and finite dimensionality;
– extension of functionals;
– weak*-compactness;
– sequence spaces and duality;
– basic properties of Banach algebras.

Suitable for: Undergraduate students Level Four

Dr Joel F. Feinstein
School of Mathematical Sciences

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.

Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

The aim of this session is to cover the basic theory of compact linear operators on Banach spaces. This includes definitions and statements of the background and main results, with illustrative examples and some proofs.

Target audience: This material is accessible to anyone who has a basic knowledge of metric space topology, and who knows what a bounded linear operator on a Banach space is. It is most likely to be suitable for postgraduate students or final year undergraduates.

This is a module framework. It can be viewed online or downloaded as a zip file.

It is as taught in 2009-2010.

This module introduces mathematical analysis building upon the experience of limits of sequences and properties of real numbers and on calculus. It includes limits and continuity of functions between Euclidean spaces, differentiation and integration.

A variety of very important new concepts are introduced by investigating the properties of numerous examples, and developing the associated theory, with a strong emphasis on rigorous proof.

This module is suitable for study at undergraduate level 2.

Dr Joel Feinstein, School of Mathematical Sciences

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.

Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

In June 2009 the Operator Algebras and Applications International Summer School was held in Lisbon. Dr Joel Feinstein taught one of the four courses available on Regularity conditions for Banach function algebras. He delivered four 90 minute lectures on and this learning object contains the slides, handouts, annotated slides and audio podcasts from each session.

Banach function algebras are complete normed algebras of bounded, continuous, complex-valued functions defined on topological spaces.

The aim of this session is to motivate students to understand why we might want to do proofs, why proofs are important, and how they can help us. In particular, the student will learn the following: proofs can help you to really see WHY a result is true; problems that are easy to state can be hard to solve (Fermat's Last Theorem); sometimes statements which appear to be intuitively obvious may turn out to be false (the Hospitals paradox); the answer to a question will often depend crucially on the definitions you are working with. Target audience: suitable for anyone with a knowledge of elementary algebra and prime numbers, as may be obtained by studying A level mathematics.