The Hahn-Banach Theorem is a fundamental result in functional analysis, a branch of mathematics. It states that if you have a linear functional defined on a subspace of a vector space, you can extend this functional to the entire space without losing its properties, such as boundedness. This extension is crucial for many applications in analysis and optimization. This theorem has significant implications in various fields, including convex analysis and topology. It helps in understanding dual spaces and provides tools for proving the existence of solutions in optimization problems. The Hahn-Banach Theorem is essential for developing the theory of Banach spaces.