Exact separation theorem for disjoint closed sets in Hilbert spaces

Using techniques of variational analysis and in terms of proximal normal cones, we establish exact separation results for finitely many disjoint closed sets in Hilbert spaces without compactness assumption, which supplement some existing fuzzy separation theorems as well as approximate projection theorems. With the help of separation results for closed sets, we provide necessary optimality conditions for optimal solutions in terms of proximal subdifferentials.