A nonlinear version of the Hahn-Banach theorem and some applications

M. Ruiz Galán

The existence of a linear functional that is bounded above by a given sublinear functional in a real vector space is guaranteed by the Hahn—Banach theorem. We first replace the role of the sublinear functional with a certain convex function whose effective domain has nonempty algebraic interior (not necessarily containing the null vector) and satisfies a condition on some of its slopes, and determine the compactness for an adequate topology of the set of the corresponding bounded above linear functionals. Then, we state a Hahn—Banach-type result by characterizing the existence of a function in a pointwise compact topological space of functions and satisfying a boundedness condition, which extends the Hahn—Banach theorem to a nonlinear framework. As a consequence, we derive some results in optimization, as a nonlinear Gordan’s theorem of the alternative.

Keywords: Hahn—Banach theorem theorems of the alternative

R. Henrion

M. J. Gisbert Francés, M. J. Cánovas Cánovas, R. Henrion, J. Parra López

F. J. Aragón Artacho, P. T. Vuong

E. Casas Rentería