Statistical Convergence In Banach Spaces

The notion of statistical convergence was introduced by Fast [9] in 1951. From then on, statistical convergence had been investigated and developed in a sequence of articles (see, for instance, [1,5,6,10,17,18,22]). Statistical convergence became an active area of research under the name of statistical convergence at the turn of the 20th century. It has appeared in a variety of topics. For example, statistical convergence has been discussed in number theory [8], trigonometric series [23], strong integral summability [4], locally convex spaces [15,18] and the structure of ideals of bounded continuous functions on locally compact spaces [3].In this paper, we introduce the notions of average convergence, μ-statistical convergence and μ-density convergence. This paper shows mainly the following results:1. Let {x_k} is a bounded sequence. Then {x_k} is statistically convergent implies {x_k} is averagely convergent;but not conversely.2. Let ∑ = 2~N(all subsets of N). Then there exists a finitely additive probability measure μ:∑→ R~+ such that:(i) For every A∈∑,then(ii) If A ∈ ∑ such that , then μ(A) = α;(iii) If {A_k} (?) ∑ with A_i∩A_j = φ for all i≠ j, and if ,3.Let...