Banach spaces with polynomial norms

A Banach space X is said to be in the class P_2nif, for all elements x and y, ∥x + ty∥²ⁿis a polynomial in real t. These spaces generalize L_2nand are precisely those Banach spaces in which linear identities can occur. We shall discuss further properties of P_2nspaces, 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 P_2nn ≤ 3. The question of embeddability is equivalent to the classical moment problem. All spaces in P_2nare 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_pp ≠ 2n, fail in the even case.