On generic uniqueness of optimal solution in an infinite dimensional linear programming problem

The class of finite dimensional linear programming problems with a unique optimal solution for a massive set of maximized functionals is described. For a weak Aspland space and a constraint set of problem that is compact in the weak topology, the uniqueness set contains a subset that is everywhere dense. It is found that for a Banach space of functions for which the set is at most countable, the unit ball is weak compact. For an arbitrary Banach space, there exists a nonzero recessive vector, such that the set can not be dense. The problem of maximizing a functional over a set shows that the set is bounded but the nonzero recessive vector does not exist. The results also show that if the massive set is not bounded then there is a functional which is not bounded from above the set.