Article ID Journal Published Year Pages File Type
4591405 Journal of Functional Analysis 2012 26 Pages PDF
Abstract

We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is . With this description we show that, for real Banach spaces X and Y, if X is a nontrivial M-ideal in Y, then (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y, it is known that (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory