Remarks on Fixed-Point Theorem, Separation of Convex Sets and Application in Linear Operator Theory

Purpose of the Study: The study aims to explore the structure of the dual space corresponding to a subspace of a reflexive Banach space equipped with a strictly convex norm. It further seeks to analyze properties of linear continuous operators, the separation of convex subsets, and the existence of invariant subspaces. Methodology: The research is based on theoretical and functional analysis techniques. It constructs the dual space under the given conditions and uses analytical methods to examine operator properties and convex set separability. Fixed-point and existence theorems are formulated using general mapping principles. Main Findings: A dual space corresponding to a subspace of a reflexive Banach space with a strictly convex norm is developed. Several properties of linear continuous operators and convex subset separation are identified. Fixed-point and existence theorems for general maps are established. Applications of this Study: The results can be applied in advanced functional analysis, particularly in operator theory, optimization problems, and mathematical modeling where the behavior of linear operators in Banach spaces is critical. These findings are also relevant to areas involving fixed-point theory and convex analysis. Novelty/Originality of this Study: This study offers a novel construction of the dual space in a specific setting of Banach spaces—reflexive with strictly convex norms—where such structures are not commonly analyzed in depth.