Let X be a Banach space. Let ?i,.(X*) the M e t space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, . . . , of the closed unit ball B ( X * ) of X* are continuous for the weak-star topology. A fundamental Hystem of norms for this space is the supremum of the absolute d u e of each element of a , . (X*) in m E ( X * ) , m = 1,2, . . . . In this paper we construct the bidual of 7-1,. (X*) when this space contains 110 copy of f l . We also show that if X is an Asplund space, then 7i,. (X*) can be represented ~8 the projective limit of a sequence of Banach spaces that are Asplund.
0. Introduction
Unless stated, all linear spaces used here throughout are assumed to be nontrivial rmd defined over the field of complex numbers. IN denotes the set of positive integers.
If X is a Banach space, X * is its conjugate and X** its second conjugate. We identify, in the usual fashion, X with a subspace of X**. B ( X ) is the closed unit ball of X. Except for special situations, the norm of any Banach space will be denoted by For a given Banach space X and a non-negative integer m, P("X) is the lin-(Bar space of all continuous mhomogeneous polynomials defined on X. We consider P ( " X ) endowed with the usual norm, i.e., for each f in this space, II . II: llfll := SUP {I f(s) I: 2 E B ( X ) l * P,,,. ( " X " ) represents the subspace of P( " X * ) formed by those polynomials whose rostrictions to B ( X * ) are continuous when the weak-star topology of X * is restricted 1.0 this ball. By P(w*) ("X*) we mean the Banach subspace of P( " X * ) algebraically rlefined by the closure of Pw= ( " X " ) in P( " X * ) when this space is endowed with the rompact open topology. 1991 Maihematicr Subjeci Claasificaiion. 46 G 20. Keywods and phmser. Infinite dimensional holomorphy, weak -star holomorphic functions.