Bridging Classical and Contemporary Duality in Locally Convex Topological Vector Spaces

This paper delves into the intricate relationship between various specialized classes of locally convex topological vector spaces and their corresponding duality theory. Building upon the foundational contributions of pioneering mathematicians in functional analysis, this work aims to provide a deeper understanding of the structural properties and interconnections within these spaces. Specifically, we explore the nuances of projective and inductive limits, analyze the characteristics of convex bornological spaces, and investigate the properties of (DF)-spaces, thereby extending classical results and offering novel perspectives on their dual representations. A significant part of this research focuses on establishing new theorems and constructing illustrative examples, particularly in the realm of nuclear spaces, to elucidate their behavior under duality mappings. This study contributes to the ongoing development of locally convex spaces by refining existing frameworks and presenting fundamental results that enhance the theoretical underpinnings of duality in infinite-dimensional analysis.