## Abstract

A Banach space X is said to be in the class P_{2n}if, for all elements x and y, ∥x + ty∥^{2n}is a polynomial in real t. These spaces generalize L_{2n}and are precisely those Banach spaces in which linear identities can occur. We shall discuss further properties of P_{2n}spaces, often in terms of the permissible polynomials p(t) = ∥x + ty∥^{2n}. For each n, the set of such polynomials forms a cone. All spaces in P_{2}are Hubert spaces. If X is a two-dimensional real space in P_{4}, then it is embeddable in L4. This is not necessarily true for spaces with more dimensions or for P_{2n}n ≤ 3. The question of embeddability is equivalent to the classical moment problem. All spaces in P_{2n}are uniformly convex and uniformly smooth and thus reflexive. They obey generally weaker versions of the Holder and Clarkson inequalities. Krivine’s inequalities, shown to determine embeddability into L_{p}p ≠ 2n, fail in the even case.

Original language | English (US) |
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Pages (from-to) | 223-235 |

Number of pages | 13 |

Journal | Pacific Journal of Mathematics |

Volume | 82 |

Issue number | 1 |

DOIs | |

State | Published - May 1979 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)