Vol. 82, No. 1, 1979

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Banach spaces with polynomial norms

Bruce Reznick

Vol. 82 (1979), No. 1, 223–235

A Banach space X is said to be in the class 𝒫2n if, for all elements x and y, x + ty2n 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 + ty2n. 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, p2n, fail in the even case.

Mathematical Subject Classification
Primary: 46B05, 46B05
Received: 14 October 1977
Published: 1 May 1979
Bruce Reznick
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