Convexity conditions for non-locally convex lattices

Abstract

First we recall that a (real) quasi-Banach space X is a complete metrizable real vector space whose topology is given by a quasi-norm satisfyingwhere C is some constant independent of x1 and x2. X is said to be p-normable (or topologically p-convex), where 0 < p ≤ l, if for some constant B we havefor any x1, …, xn, є X. A theorem of Aolci and Rolewicz (see [18]) asserts that if in C = 21/p-1, then X is p-normable. We can then equivalently re-norm X so that in (1.4) B = 1.