Uniform Boundedness Principle and Hahn-Banach Theorem for b-linear functional related to linear 2-normed space

TL;DR

This paper extends fundamental theorems like the Uniform Boundedness Principle and Hahn-Banach theorem to b-linear functionals in linear 2-normed spaces, exploring properties of these functionals and their convergence.

Contribution

It introduces the extension of classical functional analysis theorems to b-linear functionals in linear 2-normed spaces, including new concepts of continuity and weak* convergence.

Findings

Cartesian product of two 2-Banach spaces is also 2-Banach

Derived Uniform Boundedness Principle for b-linear functionals

Established Hahn-Banach extension theorem in linear 2-normed spaces

Abstract

In this paper, we will see that the Cartesian product of two 2-Banach spaces is also 2-Banach space and discuss some properties of closed linear operator in linear 2-normed space. We also describe the concept of different types of continuity of b-linear functional and derive the Uniform Boundedness Principle and Hahn-Banach extension theorem for b-linear functionals in the case of linear 2-normed spaces. We also introduce the notion of weak*convergence for the sequence of bounded b-linear functionals relative to linear 2-normed space.