On the spaces of $\lambda-$ convergent and bounded series

TL;DR

This paper introduces new sequence spaces related to $ ext{lambda}$-convergent series, establishes their properties, duals, bases, and explores matrix transformations involving these spaces.

Contribution

It defines and analyzes the properties of $cs^{ ext{lambda}}$, $cs_0^{ ext{lambda}}$, and $bs^{ ext{lambda}}$ spaces, including isomorphisms, bases, duals, and matrix class characterizations.

Findings

Spaces are linearly isomorphic to classical sequence spaces.

Constructed Schauder bases for the new spaces.

Characterized matrix classes from these spaces to classical spaces.

Abstract

The main purpose of this study is to introduce the spaces $cs^{\lambda}, cs_0^{\lambda}$ and $bs^{\lambda}$ which are $BK-$spaces of non-absolute type. We prove that these spaces are linearly isomorphic to the spaces $cs, cs_0$ and $bs$, respectively and derive some inclusion relations. Additionally, their Schauder bases have been constructed and the $\alpha-,\beta-$ and $\gamma-$ duals of these spaces have been computed. Finally, we characterize some matrix classes from the spaces $cs^{\lambda}, cs_0^{\lambda}$ and $bs^{\lambda}$ to spaces $\ell_p, c$ and $c_0$, where $1\leq p\leq\infty$ .