A multiset version of James's theorem

Let A and B be closed, convex bounded subsets of a weakly sequentially complete Banach space E which both are not compact. Then there is linear form x0⁎∈E⁎ does attain its supremum on B. In particular, given any subset A⊂E, if every x⁎∈E⁎ either attains or infimum A, then relatively The same happens for finite family but compact subsets. result only remains true in arbitrary assuming, two σ(E⁎⁎,E⁎)-cluster points E⁎⁎∖E, the fact that they generate vector subspace without nonzero vectors E, i.e.: whenx0⁎⁎∈A‾σ(E⁎⁎,E⁎)∖A y0⁎⁎∈B‾σ(E⁎⁎,E⁎)∖B verifyco({x0⁎⁎,−x0⁎⁎,y0⁎⁎,−y0⁎⁎})∩E={0}. known example shows multiset analogue Bishop-Phelps theorem true.