A New Class of Sequences Related to the lp Spaces Defined by Sequences of Orlicz Functions

By w, we denote the space of all real or complex valued sequences. If x ∈ w, then we simply write x xk instead of x xk ∞ k 1. Also, we will use the conventions that e 1, 1, . . . . Any vector subspace of w is called a sequence space. We will write l∞, c, and c0 for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by lp 1 ≤ p < ∞ , we denote the sequence space of all p-absolutely convergent series, that is, lp {x xk ∈ w : ∑∞ k 0 |xk| < ∞} for 1 ≤ p < ∞. Throughout the article, w X , l∞ X , and lp X denote, respectively, the spaces of all, bounded, and p-absolutely summable sequences with the elements in X, where X, q is a seminormed space. By θ 0, 0, . . . , we denote the zero element in X. Ps denotes the set of all subsets of N, that do not contain more than s elements. With φs , we will denote a nondecreasing sequence of positive real numbers such that s − 1 φs−1 ≤ s − 1 φs and φs → ∞, as s → ∞. The class of all the sequences φs satisfying this property is denoted by Φ. In paper 1 , the notion of λ-convergent and bounded sequences is introduced as follows: let λ λk ∞ k 0 be a strictly increasing sequence of positive reals tending to infinity, that is