A Banach space X is said to be
in the class 𝒫2n if, for all elements x and y, ∥x + ty∥2n is a polynomial in real t.
These spaces generalize L2n and are precisely those Banach spaces in which linear
identities can occur. We shall discuss further properties of 𝒫2n 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 𝒫2 are Hilbert 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 𝒫2n, n ≧ 3. The question of
embeddability is equivalent to the classical moment problem. All spaces in 𝒫2n
are uniformly convex and uniformly smooth and thus reflexive. They obey
generally weaker versions of the Hölder and Clarkson inequalities. Krivine’s
inequalities, shown to determine embeddability into Lp, p≠2n, fail in the even
case.
Department of Mathematics and Center
for Advanced Study
University of Illinois at Urbana-Champaign
1409 W. Green Street
327 Altgeld Hall
Urbana IL 61801-2975
United States
