Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars

TL;DR

This paper extends fundamental theorems of functional analysis, such as the uniform boundedness principle and Hahn-Banach theorem, to modules over hyperbolic and bicomplex scalar fields, broadening the theoretical framework.

Contribution

It introduces and proves hyperbolic and bicomplex analogues of key functional analysis theorems, expanding their applicability to new algebraic structures.

Findings

Hyperbolic and bicomplex uniform boundedness principles established

Open mapping and closed graph theorems proved for hyperbolic and bicomplex modules

Hahn-Banach separation theorem extended to these modules

Abstract

We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.