Convexity conditions for non-locally convex lattices

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 satisfying where C is some constant independent of x 1 and x 2 . X is said to be p-normable (or topologically p-convex ), where 0 < p ≤ l, if for some constant B we have for any x 1 , …, x n , є X . A theorem of Aolci and Rolewicz (see [18]) asserts that if in C = 2 1/p-1 , then X is p -normable. We can then equivalently re-norm X so that in (1.4) B = 1.