Existence of polynomials on subspaces without extension (Q2839309)

The problem of extendibility of continuous homogeneous polynomials has been addressed from many different points of view: from the perspective of the polynomial intended to be extended to any superspace or from the perspective of the Banach space from where (or to where) all the polynomials are extended.NEWLINENEWLINEFollowing the work initiated in [\textit{M. Fernández-Unzueta} and \textit{Á. Prieto}, Math. Proc. Camb. Philos. Soc. 148, No. 3, 505--518 (2010; Zbl 1197.46027)], the objective of the article under review is to provide geometric conditions on a Banach space that assure the existence of a polynomial defined on a subspace which cannot be extended to the whole space. The flavor of this study is similar to the work developed by the author (in the multilinear setting) in [Isr. J. Math. 188, 301--322 (2012; Zbl 1270.46039)]. Nevertheless, the proofs in the polynomial setting are not simple translations of the multilinear arguments. The two main results obtained here are the following: {\parindent=0.7cm\begin{itemize}\item[(1)] If a Banach space \(E\) does not have type \(p\) for any \(1<p<2\) then, for every \(n\), there exists an \(n\)-homogeneous polynomial defined on a subspace of \(E\) that cannot be extended to \(E\). \item[(2)]If \(2<k\leq n\) and \(\ell_k\) is finitely representable in \(E\), then there exists an \(n\)-homogeneous polynomial defined on a subspace of \(E\) that cannot be extended to \(E\).NEWLINENEWLINE\end{itemize}}