Some New Results on Commutative Algebra

Abstract

In this paper, we shall discuss and give applications of a polarization formula, which yields estimates of   when estimates of  are given, and estimates of the polarized forms of the terms of the Taylor expansion of a bounded holomorphic function over a bell in Banach space. The polarization formula can be used to obtain new properties of commutative algebras, with a convex topology or a convex bounded structure. The polarization formula will be proved in the first section,a  few application will be given in section 2,3,4 in which we obtain estimates of  from estimates of . We shall also define the infinite dimensional holomorphic functional calculus.