Abstract
Bishop and Phelps proved in [ l ] that every proper closed convex subset of a has many support points. (A support point of a convex set C in a real topological linear £ is a point x of C for which there is a nonzero continuous linear functional ƒ on E with f(x) =supCGCƒ(£)•) At the end of their paper they asked whether Banach space can be replaced by locally convex space in the statement of this result. In [3], Klee settled this negatively by exhibiting a proper closed convex set in R^o with no support points. The set in Klee's example is unbounded, and at the end of [3] the question was raised whether a bounded closed convex subset of a complete locally convex must have support points. (Note that a bounded closed convex subset of R*o is weakly compact and hence has support points. Indeed, if {Bi}£ml is any sequence of reflexive spaces, every bounded closed convex subset of the product H<Li^i is weakly compact and therefore has support points.) We settle Klee's question negatively:
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