A Minimax Theorem for Linear Operators

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 l(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, phi : F -> R be a continuous convex coercive map, J subset of R a compact interval and (sic) : J -> R a convex continuous function. Assume moreover that S x T has a closed range in F x F and that S is not compact. Then sup inf (phi(T-x-lambda Sx)+Psi(lambda)) = inf lambda is an element of J sup(x is an element of X) (phi(Tx-lambda Sx)+psi(lambda)). In particular, if phi is the norm of F and phi = 0, we get sup(x is an element of X) inf(lambda is an element of J) parallel to Tx-lambda Sx parallel to = inf(lambda is an element of J) sup(x is an element of X) parallel to Tx-lambda Sx parallel to.