A New Minimax Theorem for Linear Operators

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, A a finite-dimensional convex compact subset of (E pound, F), theta: F -> R be a continuous convex coercive map, and Psi: Delta -> R a convex continuous function. Assume moreover that A contains at most one compact operator. Then (sup)(x is an element of X) (inf)(T is an element of Delta) (theta(Tx) + Psi(T)) = (inf)(T is an element of Delta) (sup)(x is an element of X) (theta(Tx) + Psi(T)).