A Hahn-Banach theorem for integral polynomials

We study the problem of extendibility of polynomials over Banach spaces: when can a polynomial defined over a Banach space be extended to a polynomial over any larger Banach space? To this end, we identify all spaces of polynomials as the topological duals of a space spanned by evaluations, with Hausdorff locally convex topologies. We prove that all integral polynomials over a Banach space are extendible. Finally, we study the Aron-Berner extension of integral polynomials, and give an equivalence for non-containment of .

     

A multilinear Phelps' Lemma

We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are `close', then so are the multilinear forms.

     

A probabilistic approach to polynomial inequalities

We define a probability measure on the space of polynomials over ? ⁿ in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities.Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of ? ₁ ⁿ tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.     

Orthogonally Additive Polynomials over C(K) are Measures—A Short Proof

We give a simple proof of the fact that orthogonally additive polynomials on C(K) are represented by regular Borel measures over K. We also prove that the Aron-Berner extension preserves this class of polynomials.     

Linearization of functions

Given a space (U) of functions f:U which are continuous, we construct another space _*(U) and a map e:U *(U) linearizing all functions f (U) (i.e. there are L_{f *}(U) such that L_f^e=f). Such linearizations are stronger than mere preduals for (U), for example for (U)=₁, linearizations correspond to preduals of ₁ which are isomorphic to c₀. We also address the vector-valued case. A number of such linearizing constructions are to be found in the literarture, mostly for certain spaces of holomorphic functions. The procedure presented here generalizes all these special cases.Mathematics Subject Classification (2000):46E10, 46G20     

Zeros of Polynomials on Banach Spaces: The Real Story

Let E be a real Banach space. We show that either E admits a positive definite 2-homogeneous polynomial or every 2-homogeneous polynomial on E has an infinite dimensional subspace on which it is identically zero. Under addition assumptions, we show that such subspaces are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homogeneous polynomials and give necessary and sufficient conditions on a compact set K so that C(K) admits a positive definite 2-homogeneous polynomial or a positive definite nuclear 2-homogeneous polynomial.     

To what extent does the dual Banach spaceE′ determine the polynomials overE?

We show that under conditions of regularity, ifE′ is isomorphic toF′, then the spaces of homogeneous polynomials overE andF are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.