FRECHET SPACES OF HOLOMORPHIC-FUNCTIONS WITHOUT COPIES OF L(1)

Let X be a Banach space. Let H-w(X*) the Frechet space whose elements are the holomorphic functions defined on X whose restrictions to eac h multiple mB(X), m = 1,2,..., of the closed unit ball B(X*) of X* ar e continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of Hw(X*) in mB(X*), m = 1,2,.... In this paper we construct the bidu al of Hw(X*) when this space contains no copy of l(1). We also show t hat if X is an Asplund space, then Hw(X*) can be represented as the p rojective limit of a sequence of Banach spaces that are Asplund.