A Minimax Theorem for Linear Operators
Keywords:
Minimax, Banach spaces, linear operators.Abstract
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 ℓ1, 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, ϕ : F → R be a continuous convex coercive map, J ⊂ R a compact interval and ψ : J → R a convex continuous function. Assume moreover that S ×T has a closed range in F ×F and that S is not compact. Then
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