The main objective of this chapter is to present the separation theorems, important consequences of Hahn-Theorem theorem. Therefore, we begin with an overview on convex sets and convex functionals. Then go on with the Hahn-Banach theorem and separation theorems. Follow these results specification: first for normed spaces and then for a subclass of these spaces, the Hilbert spaces. In this last case plays a key role the Riesz representation theorem. Separation theorems are key results in convex programming. Then the chapter ends with the outline of applications of these results in convex programming, Kuhn-Tucker theorem, and in minimax theorem, two important tools in operations research, management and economics,for instance.

This work is financed by national funds through FCT - Fundação para a Ciência e Tecnologia, I.P., under the project UID/Multi/04466/2019. Furthermore, I would like to thank the Instituto Universitário de Lisboa and ISTAR-IUL for their support.

Hahn-Banach theorem,Separation theorems,Convex programming,Kuhn-Tucker theorem,Minimax theorem

• Mathematics - Natural Sciences
• Computer and Information Sciences - Natural Sciences