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