We review the feasibility conditions for the Hilbert space environment and study the feasibility cone \(F+P\) . The ensuing developments are an important component of the theory of polyhedra from the range space point of view and will be used to solve linear optimization problems over polyhedra. In addition we will apply this theory to derive a further result on the closure of pointed cones, which also illustrates the mechanism, by which a pointed non-closed cone can have a “sparsity” which allows its closure to be dense. Such sparsity is never afforded by closed pointed cones.