Neutrosophic Quadruple Metric Spaces and Neutrosophic Quadruple Normed Spaces

Abstract

This study establishes fundamental results in the emerging domains of neutrosophic 
quadruple metric spaces and neutrosophic quadruple normed spaces. Building upon the recent 
definition of neutrosophic quadruple metric spaces, the first primary contribution is the 
development of fixed-point theory within this generalized framework. Specifically, we formulate 
and rigorously prove an analogue of the Banach contraction principle tailored for neutrosophic 
quadruple metric spaces. Furthermore, we establish additional fixed-point theorems applicable in 
this context, extending foundational results from classical metric spaces and simpler neutrosophic 
structures to handle the increased complexity and uncertainty modeled by quadruple-valued 
neutrosophic sets. The second major contribution involves the algebraic generalization of normed 
spaces. We introduce the novel concept of neutrosophic quadruple normed spaces. Within these 
spaces, we define an appropriate norm structure capable of measuring the "magnitude" of vectors 
characterized by quadruple-valued neutrosophic components. We systematically investigate and 
establish various fundamental properties of this newly defined norm, exploring concepts such as 
statistical convergence, Cauchy statistical convergence, boundedness, and continuity within the 
neutrosophic quadruple normed spaces. Additionally, we address the specific case of neutrosophic 
quadruple vector spaces, defining and analyzing the corresponding norm structure in this 
specialized context. The results generalize and extend prior work in fuzzy, intuitionistic fuzzy, and 
standard neutrosophic metric and normed spaces. 

DOI 10.5281/zenodo.17969819