Weak and weak* $I^K$-convergence in normed spaces

TL;DR

This paper introduces and studies weak and weak* $I^K$-convergence concepts in normed spaces, generalizing existing convergence notions by incorporating two ideals, and explores their properties and limit points.

Contribution

It presents a new framework for weak and weak* $I^K$-convergence in normed spaces, extending prior convergence concepts with the use of two ideals.

Findings

Defined weak and weak* $I^K$-convergence in normed spaces.

Analyzed properties of weak $I^K$ and weak* $I^K$-limit points.

Established relationships between these convergence types.

Abstract

The main object of this paper is to study the concept of weak $I^K$-convergence, a generalization of weak $I^*$-convergence of sequences in a normed space, introducing the idea of weak* $I^K$-convergence of sequences of functionals where $I,K$ are two ideals on $\mathbb{N}$, the set of all positive integers. Also we have studied the ideas of weak $I^K$ and weak* $I^K$-limit points to investigate the properties in the same space.