Existence of polynomials on subspaces without extension

Fernández-Unzueta, Maite

doi:10.1090/S0002-9939-2013-11703-X

Existence of polynomials on subspaces without extension
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by Maite Fernández-Unzueta PDF

Proc. Amer. Math. Soc. 141 (2013), 2389-2399 Request permission

Abstract:

We prove the existence of a polynomial of degree $d$ defined on a closed subspace that cannot be extended to the Banach space $E$ (in particular, the existence of a nonextendible polynomial) in the following cases: (1) $d\geq 2$ and $E$ does not have type $p$ for some $1<p<2$; (2) the space $\ell _k$, $k\in \mathbb {N}$, $2<k\leq d$, is finitely representable in $E$. In each of these cases we prove, equivalently, the existence of a closed subspace $F\subset E$ such that the subspace $\hat {\otimes }^{d}_{s,\pi }{F}$ is not closed in $\hat {\otimes }^{d}_{s,\pi }{E}$.

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