Investigating Linear 2-Normed and 2-Inner Product Spaces: Orthonormal Sets and Their Applications in Analysis

Abstract

This study provides a comprehensive analysis of linear 2-normed spaces and 2-inner product spaces, building on foundational theories and previous research. We clarify key definitions, structures, and properties governing these mathematical constructs, demonstrating their relevance to functional analysis and approximation theory. Our investigation highlights the significance of best approximations and their connections to optimization problems, as well as the crucial roles of completeness and convergence in influencing the behavior of sequences and functions. The insights gained from this work not only enhance current understanding but also open avenues for future research, encouraging mathematicians to explore unresolved questions and refine existing theories. Overall, this research underscores the importance of linear 2-normed and 2-inner product spaces within the broader mathematical landscape.