SOME PROPERTIES OF SEQUENCE SPACE _ BV (f; p; q; s)

Abstract

Let `1 and c denote the Banach spaces of real bounded and convergent sequencesx = (xn) normed by kxk = supnjxnj ; respectively.Let be a one to one mapping of the set of positive integers into itself such thatk(n) = k1(n); k = 1; 2; ::: .A continuous linear functional ' on `1 is saidto be an invariant mean or a mean if and only if(i) ' (x) 0 when xn 0 for all n;(ii) ' (e) = 1; where e = (1; 1; 1; :::) and(iii) 'x(n) = ' (fxng) for all x 2 `1:If is the translation mapping n ! n + 1; a mean is often called a Banachlimit [3], and V is the set of convergent sequences, that is, the set of boundedsequences all of whose invariant means are equal, is the set ^f of almost convergentsequences