Some results on quasi-convergence in gradual normed linear spaces

Abstract

Summability theory and convergence of sequence in various spaces have become significant topics in mathematical analysis. Researchers have presented some elegant works about gradual normed linear spaces in literature. This chapter intends to survey some recent developments of convergence of double real number sequences in gradual normed linear spaces and its applications. In this chapter, we set forth with the new notions of quasi-invariant convergence and quasi-invariant statistical convergence of double sequences in the gradual normed linear spaces, and we examine a characterization of a bounded sequence to be quasi-invariant convergent. Also, we investigate the notions of statistical lacunary summability and strongly https://www.w3.org/1998/Math/MathML”> ? q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003460169/3e12725d-9588-47fc-96d1-4ce3f69e6c55/content/C010_equ_0001.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> convergence for double sequences in gradual normed linear spaces. By utilizing four-dimensional conservative matrices, we demonstrate an inequality related to the notion of statistical lacunary limit inferior (superior) of real bounded double sequences in gradual normed linear spaces. In addition, we introduce the concepts of quasi-almost convergence and quasi-almost statistical convergence of double sequences in gradual normed linear spaces. Afterward, we give the notions of quasi-strongly almost convergence and quasi https://www.w3.org/1998/Math/MathML”> q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003460169/3e12725d-9588-47fc-96d1-4ce3f69e6c55/content/C010_equ_0002.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> -strongly almost convergence of double sequences and study the relationship among these concepts. The whole research work is done in more of a theoretical direction. Theorems are proved in the light of gradual normed linear spaces theory approach. Results are obtained through different perspectives, and new examples are produced to justify the counterparts and demonstrate the existence of introduced notions. The results established in this research work supply an exhaustive foundation in gradual normed linear spaces and make a significant contribution to the theoretical development of gradual normed linear spaces in the literature. The conclusions of this chapter are expected to be a source for statistics and mathematics researchers in the areas of convergence methods for sequences and implementations in gradual normed linear spaces. © 2023 Elsevier B.V., All rights reserved.