After presenting some structural notions on Hilbert spaces, which constitute fundamental support for this work, we approach the goals of thechapter. First,studyaboutconvexsets,projections,andorthogonality, whereweapproachtheoptimizationprobleminHilbertspaceswithsome generality. Then the approach to Riesz 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. These theorems are very important in the applications. Moreover, the presented strict separation theorems and the Riesz representation theorem have 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 Universit´ario de Lisboa and ISTAR-IUL for their support.

Hilbert spaces,Convex sets,Projections,Orthogonality,Riesz representation theorem,Kuhn-Tucker theorem,Minimax theorem

• Mathematics - Natural Sciences
• Physical Sciences - Natural Sciences