Representation of Uniform Boundedness Principle and Hahn-Banach Theorem in linear n-normed space

TL;DR

This paper extends fundamental functional analysis principles, specifically the Uniform Boundedness Principle and Hahn-Banach Theorem, to linear n-normed spaces using bounded b-linear functionals, and explores their properties and applications.

Contribution

It introduces the concepts of b-linear functionals and their continuity, and generalizes key theorems to linear n-normed spaces, including weak* convergence.

Findings

Established the Uniform Boundedness Principle in linear n-normed spaces.

Proved the Hahn-Banach extension Theorem for bounded b-linear functionals.

Discussed applications and introduced weak* convergence in this context.

Abstract

The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.