Some properties of sequence space BV (f; p; q; s)

Abstract

Let `1 and c denote the Banach spaces of real bounded and convergent sequences x = (xn) normed by kxk = sup n jxnj ; respectively. Let be a one to one mapping of the set of positive integers into itself such that k (n) = k1 (n) ; k = 1; 2; ::: .A continuous linear functional ' on `1 is said to 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 Banach limit [3], and V is the set of convergent sequences, that is, the set of bounded sequences all of whose invariant means are equal, is the set ^f of almost convergent sequences