On a class of reflexive spaces related to Ulam's conjecture on measurable cardinals.
A Banach space is called -reflexive if for any cover of by weakly open sets there is a finite subfamily covering some ball of radius 1 centered at a point with . We prove that an infinite-dimensional separable Banach space is -reflexive (-reflexive for some ) if and only if each -net for has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of . We show that the quasireflexive James space is -reflexive for no . We do not know...