Summability is an important concept in sequence spaces. One summability strongly Cesaro summable. In this paper, we study a subset of the set all vector-valued 2-modular space. Some facts that investigated paper include linearity, existence modular and completeness with respect to these modular.

the object of this paper is to establish a summability factor theorem for summabilitya, , k  1 k  where a is the lower triangular matrix with non-negative entries satisfying certain conditions.

we obtain sufficient conditions for the series σanλn to be absolutely summable of order k by atriangular matrix.

in this paper, the notion of n, pn - summability to generalize the concept of statisticalconvergence is used. we call this new method weighted statistically convergence. we also establish itsrelationship with statistical convergence, c,1-summability and strong   n n, p -summability.

It is well known that (see [9]) Cesaro means of 2π-periodic functions f ∈ Lp(T) (1 ≤ p ≤ ∞) converges by norms. Hereby T is denoted the interval (−π,π). The problem of the mean summability in weighted Lebesgue spaces has been investigated in [6]. A 2π-periodic nonnegative integrable function w : T→R1 is called a weight function. In the sequel by L p w(T), we denote the Banach function space of ...

In this paper, we will introduce the notion of convergence two dimensional interval sequences and show that set all numbers is a metric space. Also, some ordinary vector norms be extended to vectors. Furthermore, give definitions statistical convergence, statistically Cauchy Cesaro summability for get relationships between them.