A Banach space X is said to be in the class P2nif, for all elements x and y, ∥x + ty∥2nis a polynomial in real t. These spaces generalize L2nand are precisely those Banach spaces in which linear identities can occur. We shall discuss further properties of P2nspaces, 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 P2are Hubert spaces. If X is a two-dimensional real space in P4, then it is embeddable in L4. This is not necessarily true for spaces with more dimensions or for P2nn ≤ 3. The question of embeddability is equivalent to the classical moment problem. All spaces in P2nare 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 Lpp ≠ 2n, fail in the even case.
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