Some Sequence Spaces and Statistical Convergence

for sets of sequences x = (xk) which are strongly (V ,λ)-summable to L, that is, xk → L[V,λ]. We recall that a modulus f is a function from [0,∞) to [0,∞) such that (i) f(x)= 0 if and only if x = 0; (ii) f(x+y)≤ f(x)+f(y) for all x, y ≥ 0; (iii) f is increasing; (iv) f is continuous from the right at 0. It follows that f must be continuous on [0,∞). A modulus may be bounded or unbounded. Maddox [6] and Ruckle [9] used the modulus f to construct sequence spaces. In this paper, we introduce the strongly (V ,λ)-convergent sequences and give the relation between strongly (V ,λ)-convergence and strongly (V ,λ)-convergence with respect to a modulus.