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Minimax Theory and its Applications 03 (2018), No. 1, 131--160
Copyright Heldermann Verlag 2018

A New Minimax Theorem for Linear Operators

Jean Saint Raymond
Institut de Mathématique de Jussieu, Université P. et M. Curie, 4 place Jussieu, 75252 Paris 05, France


The aim of this note is to prove the following minimax theorem which generalizes a result by B. Ricceri and extends a previous result of the author: let $E$ be a infinite-dimensional Banach space, $F$ be a Banach space, $X$ be a convex subset of $E$ whose interior is non-empty for the weak topology on bounded sets, $\Delta$ a finite-dimensional convex compact subset of $\mathcal{L}(E,F)$, $\varphi\colon F\to \mathbb{R}$ be a continuous convex coercive map, and $\psi\colon \Delta\to \mathbb{R}$ a convex continuous function. Assume moreover that $\Delta$ contains at most one compact operator. Then $$ \sup_{x\in X}\inf_{T\in \Delta}\bigl( \varphi( Tx)+\psi(T)\bigr) = \inf_{T\in \Delta}\sup_{x\in X}\bigl( \varphi( Tx)+\psi(T)\bigr)\ . $$

Keywords: Minimax, Banach spaces, linear operators.

MSC: 49J35, 46B04, 46B50

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