The main objective of this article is to present the separation theorems, important consequences of the Hahn-Theorem theorem. These results are the fundamentals of the convex optimization mathematical approach. Along this work begin to consider vector spaces, in general, then normed spaces and posteriorly Hilbert spaces. It ends with the presentation of applications of these results in convex programming and in minimax theorem, two important tools in operations research, management and economics, for instance.

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