(untitled) (International Journal of Food and Nutritional Sciences, 2022)
The Hahn-Banach theorem is one of the major theorems that we face in a first course on Functional Analysis. We have a very powerful collection of theorems with the Banach-Steinhaus theorem, the open mapping theorem, and the closed graph theorem when we combine them. The third, fourth, and fifth theorems all need the completeness of the spaces involved, whereas the Hahn-Banach theorem does not. It’s available in two styles: analytic and geometric. The geometric version deals with the separation of disjoint convex sets using hyperplanes, whereas the analytic version deals with the extension of continuous linear functionals from a subspace to the full space with predefined properties. Both versions have applications outside of Functional Analysis, such as in optimization theory and partial differential equations theory, to mention a few. The Hahn-Banach theorem and some of its applications will be discussed in this article. We will not provide extensive proofs of the main theorems because they are available in any functional analysis book