Existence and Uniqueness of Vector of Minimum Norm in Real Hilbert Space

Student: Dorathy Chiyenkem Dominic (Project, 2025)
Department of Mathematics and Statistics
University of Port-Harcourt, Rivers State

Abstract

\chapter*{ABSTRACT}
This study investigates the existence and uniqueness of vector of minimum norm within closed convex subsets of real Hilbert spaces, a foundational concept in functional analysis and optimization. Through rigorous mathematical proofs, we establish that for any non-empty closed convex set \( C \subset H \), where \( H \) is a real Hilbert space, there exists a unique vector \( x_0 \in C \) that minimizes the norm \( \|x_0\| \) over all elements in \( C \). The existence proof relies on the completeness of \( H \) and the closed nature of \( C \), which ensure that a Cauchy sequence within \( C \) converges to a limit point in \( C \) with minimum norm. The uniqueness is demonstrated using the convexity of \( C \), guaranteeing that no two distinct points in \( C \) can attain the same minimum norm. These results have significant implications for optimization, as they form the basis for projection techniques, least squares approximation, and error minimization methods across diverse fields such as data science, physics, and engineering. The findings underscore the importance of Hilbert spaces in facilitating structured and efficient solutions to complex analytical problems, offering a robust mathematical framework for theoretical advancements and practical applications alike.

Keywords
Hilbert space minimum norm vector projection theorem functional analysis convex analysis optimization theory Riesz representation theorem orthogonality existence theorem uniqueness theorem
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