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This article makes no claim to originality, other than, perhaps, the simplestatement here called the {\it Abstract Maximum Principle}. Actually, the wholecontents are strongly based on some H. Sussmann's and coauthors' papers, inwhich, in a much more general context, the set-separation approach is regardedas foundational for necessary conditions for minima. So, rather than being theexposition of original material, this paper has mainly a pedagogical purpose.From the Abstract Maximum Principle it is possible to deduce several necessaryconditions for both finite dimensional minimum problems and for optimal controlproblems. More in general, this Principle seems apt to capture someconsequences of the geometric and topological idea of (possibly vector-valued)minimization in a parametrized problem.

An Abstract Maximum Principle for constrained minimum problems