Advanced Mathematics

Dr Joel Feinstein's talk at Banach Algebras 2011, Waterloo, Ontario, Canada. He discussed joint work with Herb Kamowitz on compact, power compact, quasicompact and Riesz endomorphisms of commutative Banach algebras, along with some background, examples, and questions.

Dr Joel Feinstein's talk at Banach Algebras 2011, Waterloo, Ontario, Canada. He discussed joint work with Herb Kamowitz on compact, power compact, quasicompact and Riesz endomorphisms of commutative Banach algebras, along with some background, examples, and questions.

This is a lecture from Dr Feinstein's 4th-year module G14FUN Functional Analysis. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ and, in particular, the Functional Analysis screencasts blog page at http://wp.me/PosHB-8v In this screencast, Dr Feinstein discusses two famous results concerning collections of bounded linear operators, one of which is a corollary of the other. Both of these results have been called the Banach-Steinhaus Theorem (by various authors). The stronger of these two results is the one which is also known as the Uniform Boundedness Principle. This material is suitable for those with a good background knowledge of metric spaces and normed spaces. In particular, the student should know about bounded (continuous) linear operators between normed spaces, and the Baire Category Theorem for complete metric spaces.

This is a lecture from Dr Feinstein's 4th-year module G14FUN Functional Analysis. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ and, in particular, the Functional Analysis screencasts blog page at http://wp.me/PosHB-8v In this screencast, Dr Feinstein introduces the weak topology on a normed space and the weak star topology on the dual space. He then proves the Banach-Alaoglu theorem, that the closed unit ball of the dual space is weak star compact. This material is suitable for those with a basic knowledge of normed spaces and their duals, and of infinite products of topological spaces, including Tychonoff's theorem on arbitrary products of compact topological spaces.

This is a lecture from Dr Feinstein's 4th-year module G14FUN Functional Analysis. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ and, in particular, the associated blog post at http://wp.me/posHB-7y In this screencast, Dr Feinstein proves the Baire Category Theorem for complete metric spaces - a countable intersection of dense, open subsets of a complete metric space must be dense. This material is suitable for those with a knowledge of metric space topology and, in particular, dense subsets and complete metrics.