Applications of Ultraproducts to Infinite Dimensional Holomorphy.

  • Lindström M
  • Ryan R
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The purpose of this paper is to apply ultraproduct techniques to someproblems in infinite dimensional holomorphy. The central problem we consideris the following: Given a continuous polynomial P, or more generally, aholomorphic function, f, defined on a Banach space X, can we extend Por f to a larger space containing X? Questions such as these were firsttackled by {\it R. M. Aron} and {\it P. D. Berner} [Bull. Soc. Math. France106, 3-24 (1978; Zbl 0378.46043)]. They showed how polynomials, and certainholomorphic functions, can be extended to the bidual X^{**}. From this,they were able to construct extensions for other spaces containing X.However, some questions were left open. For example, it was not knownwhether the Aron-Berner extension of a continuous polynomial P from X toX^{**} had the same norm as P. This question was recently answered inthe affirmative by {\it A. M. Davie} and {\it T. W. Gamelin} [Proc. Am.Math. Soc. 106, No. 2, 351-356 (1989; Zbl 0683.46037)]. \par We present anew approach to this extension problem. Our approach is to work with anultrapower (X)_u of the Banach space X rather than the bidual of X.There is a canonical embedding of X into (X)_u, and it is relativelysimple to construct extensions of polynomials and holomorphic functions fromX into (X)_u. For certain special ultrapowers of X we have roughlyspeaking, X\subset X^{**} \subset (X)_u, and so we obtain extensions fromX to its bidual as byproduct of our extension process. There is not one,but several ultrapower extension processes. One of these processes ismodelled on the Aron-Berner method, and in this case we extend the scope ofthe result of Davie and Gamelin mentioned above. The other extension processwhich we discuss is more adaptable for dealing with holomorphic functions.\par Our methods yield new results concerning the polarization constants ofa Banach space. The polarization constants of X are a sequence of realnumbers K_n (X) which contain information about the geometric structure ofX. The number K_n (X) arises when one compares the norm of a homogeneouspolynomial of degree n on X with the norm of the symmetric n-linearfunction which generates the polynomial. We show that the bidual X^{**}has the same polarization constants as X, at least when the bidual has themetric approximation property.




Lindström, M., & Ryan, R. A. (1992). Applications of Ultraproducts to Infinite Dimensional Holomorphy. MATHEMATICA SCANDINAVICA, 71, 229.

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