After presenting some structural notions on Hilbert spaces, which constitute a fundamental support for this work, we approach the goals of the chapter. First, a study about convex sets, projections and orthogonality, where we approach the optimization problem in Hilbert spaces with some generality. Then the approach to Riez representation theorem in this field, important in the rephrasing of the separation theorems. Then we give a look to the strict separation theorems as well as to the main results of convex programming: Kuhn-Tucker theorem and minimax theorem. Both these theorems are very important in the applications. Moreover, the strict separation theorems presented and the Riez representation theorem have a key importance in the demonstrations of Kuhn-Tucker and minimax theorems and respective corollaries.

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.

convex sets,Hilbert spaces,Kuhn-Tucker Theorem,minimax theorem,Orthogonality,projections,Riez representation theorem

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