Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Minimax Theory and its Applications 03 (2018), No. 1, 131--160Copyright 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 jean.saint-raymond@imj-prg.fr [Abstract-pdf] 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 [ Fulltext-pdf  (236  KB)] for subscribers only.