A minimax theorem for functions taking values in a Riesz space
Let \((E,C)\) be an order-complete topological Riesz space, where \(C\) is the positive cone. The objective of the paper is to formulate the following Riesz space version of a Hahn-Banach type theorem proved by \textit{B. Fuchssteiner} and \textit{H. König} for real functions [General inequalities 2, Proc. 2nd int. Conf., Oberwolfach 1978, ISNM Vol. 47, 255--266 (1980; Zbl 0452.46002)]: Let \(F\) be a real vector space, \(\vartheta\colon F\to (E,C)\) be a sublinear operator and let \(G\) be a nonempty subset of \(F\) verifying some suitable assumptions of `convexity' type. If \(\vartheta|_G\geq 0\), then there is a linear operators \(\phi\colon F\to E\) such that \(\phi\leq \vartheta\) and \(\vartheta|_G\geq 0\). Then as a consequence of this theorem, some minimax theorems in Riesz space are given, under very general convexity assumptions.
scientific article; zbMATH DE number 4062006