About the Banach Envelope of l_1,∞

Abstract

We study the Banach envelope of the quasi-Banach space l_1,∞ consisting of all sequences x=(ξ _k) with s_n(x)=O(1/n), where (s_n (x)) denotes the non-increasing rearrangement of x=(ξ _k) The situation turns out to be much more complicated than that in the well-known case of the separable subspace l^o_1,∞, whose members are characterized by s_n(x) =o(1/n).