A multiset version of James's theorem

Abstract

Let A and B be closed, convex and bounded subsets of a weakly sequentially complete Banach space E which both are not weakly compact. Then there is a linear form x0⁎∈E⁎ which does not attain its supremum on A and on B. In particular, given any bounded subset A⊂E, if every x⁎∈E⁎ either attains its supremum or infimum on A, then A is weakly relatively compact. The same happens for a finite family of closed, convex bounded but not weakly compact subsets. The result only remains true in arbitrary Banach space assuming, for any two σ(E⁎⁎,E⁎)-cluster points of A and B in E⁎⁎∖E, the fact that they generate vector subspace without nonzero vectors of E, i.e.: when x0⁎⁎∈A‾σ(E⁎⁎,E⁎)∖A and y0⁎⁎∈B‾σ(E⁎⁎,E⁎)∖B verify co({x0⁎⁎,−x0⁎⁎,y0⁎⁎,−y0⁎⁎})∩E={0}. A known example shows that a multiset analogue for the Bishop-Phelps theorem is not true.