Comprehensive Characterization of the Finest Locally Convex Topology with Applications

Abstract

Elvin Rada*

This paper provides a comprehensive characterization of the finest locally convex topology S on an infinite-dimensional real vector space E. By identifying E with the algebraic direct sum L i∈I R via a Hamel basis, we establish that S coincides with the locally convex direct sum topology. We derive some core structural properties of S, including its duality, boundedness, and topological decomposition behavior. Furthermore, we show that for any subset C ⊂E, compactness, total boundedness with completeness, and boundedness with closedness are equivalent. These results resolve open questions in the theory of locally convex spaces and provide new insights into the structural properties of infinite-dimensional vector spaces. In the end we highlight two significant applications in duality theory and functional analysis.

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