Applications of Ultraproducts to Infinite Dimensional Holomorphy.

The purpose of this paper is to apply ultraproduct techniques to some\nproblems in infinite dimensional holomorphy. The central problem we consider\nis the following: Given a continuous polynomial P, or more generally, a\nholomorphic function, f, defined on a Banach space X, can we extend P\nor f to a larger space containing X? Questions such as these were first\ntackled by {\it R. M. Aron} and {\it P. D. Berner} [Bull. Soc. Math. France\n106, 3-24 (1978; Zbl 0378.46043)]. They showed how polynomials, and certain\nholomorphic functions, can be extended to the bidual X^{**}. From this,\nthey were able to construct extensions for other spaces containing X.\nHowever, some questions were left open. For example, it was not known\nwhether the Aron-Berner extension of a continuous polynomial P from X to\nX^{**} had the same norm as P. This question was recently answered in\nthe affirmative by {\it A. M. Davie} and {\it T. W. Gamelin} [Proc. Am.\nMath. Soc. 106, No. 2, 351-356 (1989; Zbl 0683.46037)]. \par We present a\nnew approach to this extension problem. Our approach is to work with an\nultrapower (X)_u of the Banach space X rather than the bidual of X.\nThere is a canonical embedding of X into (X)_u, and it is relatively\nsimple to construct extensions of polynomials and holomorphic functions from\nX into (X)_u. For certain special ultrapowers of X we have roughly\nspeaking, X\subset X^{**} \subset (X)_u, and so we obtain extensions from\nX to its bidual as byproduct of our extension process. There is not one,\nbut several ultrapower extension processes. One of these processes is\nmodelled on the Aron-Berner method, and in this case we extend the scope of\nthe result of Davie and Gamelin mentioned above. The other extension process\nwhich we discuss is more adaptable for dealing with holomorphic functions.\n\par Our methods yield new results concerning the polarization constants of\na Banach space. The polarization constants of X are a sequence of real\nnumbers K_n (X) which contain information about the geometric structure of\nX. The number K_n (X) arises when one compares the norm of a homogeneous\npolynomial of degree n on X with the norm of the symmetric n-linear\nfunction which generates the polynomial. We show that the bidual X^{**}\nhas the same polarization constants as X, at least when the bidual has the\nmetric approximation property.