Existence of polynomials on subspaces without extension

Abstract

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