Claremont Colleges Analysis Seminar
2008-2009

Title: Normed algebras of differentiable functions on compact plane sets

Abstract:  Let K be a compact subset of the plane. It is very well known that C(K), the linear space of all continuous functions on K is a Banach algebra for the pointwise product and the uniform norm, | . |_K. Several closed subalgebras of C(K), called P(K), R(K), and A(K) have been much studied. I shall recall their definitions and properties, which depend on the nature of the set K.

In the case where K is perfect, there is a very natural concept of f being continuously differentiable on K, with derivative f '. Now the natural norm on the space of these functions is || f || = |f|_K +|f '|_K, and in this way we obtain a normed algebra. But now our algebra is not necessarily complete. We shall discuss when it is complete - and what its completion is when it is not complete - in terms of the geometry of K.

This talk should be accessible to students who know what a normed space is. The talk is based on joint work with Joel Feinstein of Nottingham.