Vol. 40, No. 3, 1972

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Maximinimax, minimax, and antiminimax theorems and a result of R. C. James

Stephen Simons

Vol. 40 (1972), No. 3, 709–718
Abstract

This paper contains a number of minimax theorems in various topological and non-topological situations. Probably the most interesting is the following: if X is a nonempty bounded convex subset of a real Hausdorff locally convex space E with dual Eand each φ Eattains its supremum on X then

     (                                         )
{ for all nonempty convex equicontinuous Y ⊂ E′}
(∗)                   infsup⟨X,y⟩ ≦ supinf⟨x,Y ⟩  .
(                y∈Y           x∈X        )

It is also proved that if () is true and X is complete then X is w(E,E)-compact. Combining these results, a proof of a well known result of R. C. James is obtained.

Mathematical Subject Classification
Primary: 46A05
Milestones
Received: 28 January 1971
Published: 1 March 1972
Authors
Stephen Simons