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Minimax Theory and its Applications 01 (2016), No. 2, 291--305
Copyright Heldermann Verlag 2016

A Minimax Theorem for Linear Operators

Jean Saint Raymond
Université P. et M. Curie, Institut de Mathématique, Boîte 186 - 4 place Jussieu, 75252 Paris Cedex 05, France


\newcommand\R{\mathbb R} \let\ph=\varphi The aim of this note is to prove the following minimax theorem which generalizes a result by B. Ricceri: let $E$ be an infinite-dimensional Banach space not containing $\ell^1$, $F$ be a Banach space, $X$ be a convex subset of $E$ whose interior is non-empty for the weak topology on bounded sets, $S$ and $T$ be linear and continuous operators from $E$ to $F$, $\ph : F\to \R$ be a continuous convex coercive map, $J\subset \R$ a compact interval and $\psi : J\to \R$ a convex continuous function. Assume moreover that $S\times T$ has a closed range in $F\times F$ and that $S$ is not compact. Then $$ \sup_{x\in X}\inf_{\lambda\in J}\bigl( \ph( Tx-\lambda Sx)+ \psi(\lambda)\bigr) = \inf_{\lambda\in J} \sup_{x\in X} \bigl(\ph( Tx-\lambda Sx)+\psi(\lambda)\bigr)\,. $$ In particular, if $\ph$ is the norm of $F$ and $\psi=0$, we get $$ \sup_{x\in X} \inf_{ \lambda\in J} \lVert Tx-\lambda Sx\rVert = \inf_{ \lambda\in J}\sup_{x\in X} \lVert Tx-\lambda Sx\rVert\,. $$

Keywords: Minimax, Banach spaces, linear operators.

MSC: 49J35, 46B04, 46B50

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