Polynomial approximation in Banach spaces

We define new classes of topological algebras of holomorphic functions on open subsets of Banach spaces, and on open subsets of dual Banach spaces. We investigate properties and derive results concerning polynomial approximation on such algebras. We give an explicit description of their spectra, derive results on finitely generated ideals, and theorems of Banach–Stone type. We show that under certain conditions on the open subset U, this new algebra coincides with Hwu(U), deriving new results on Hwu(U).