Concrete semispaces and lexicographic separation of convex sets
Calvin C. Moore · Pacific Journal of Mathematics · 1973
raised a question concerning lexicographic separation of disjoint closed convex sets in a locally convex space by means of a semispace with a representation utilizing continuous linear functionals.This question is answered and related results involving hyperplane separation and reflexivity in Banach spaces are discussed.1* Introduction* If p is a vector in a real linear space E, then a semispace at p is a maximal convex subset of E ~ {p}.This notion was introduced by Hammer [3] in 1955 and the structure of semispaces was determined by Klee [7] in 1956.The present paper is strongly dependent upon this work of Klee.The concept of a concrete semispace in a locally convex space is introduced and a basic representation theorem for such an object is proved.Concrete separation (lexicographic separation via concrete semispaces) is defined and a criterion is given for determining when two sets are not concretely separated.Disjoint closed convex subsets A and B of a pre-Hilbert space E such that A -B is dense in E and A -B has nonempty core are exhibited.These sets cannot be separated concretely and this implies a negative answer to Klee's question concerning concrete separation of disjoint closed convex sets in a locally convex space.Hyperplane separation and concrete separation are contrasted.It is proved that a Banach space E is reflexive if and only if each disjoint pair of bounded closed convex subsets of E is concretely separated.