Completions of normed algebras of differentiable functions
We look at normed spaces of differentiable functions on compact plane sets, including the spaces of infinitely differentiable functions considered by Dales and Davie in . For many compact plane sets the classical definitions give rise to incomplete spaces. We introduce an alternative definition of differentiability which allows us to describe the completions of these spaces. We also consider some associated problems of polynomial and rational approximation.