Hyperspaces of Topological Vector Spaces: Their Embedding in Topological Vector Spaces

Let L be a real (Hausdorff) topological vector space. The space %[L] of nonempty compact subsets of L forms a (Hausdorff) topological semivector space with singleton origin when 3C[L] is given the uniform (equivalently, the finite) hyperspace topology determined by L. Then %[L] is locally compact iff L is so. Furthermore, 90.[Z,], the set of nonempty compact convex subsets of L, is the largest pointwise convex subset of %[L] and is a cancellative topological semivector space. For any nonempty compact and convex set X C L, the collection SCSlX] C CK2[L] is nonempty compact and convex. L is iseomorphically embeddable in 9©[L] and, in turn, there is a smallest vector space £ in which 3 0). For any set X, [X] denotes the set of nonempty subsets of X. When A is a topological space, %[X] denotes the set of compact nonempty subsets of X. When X lies in a real vector space, S[A] denotes the set of convex nonempty subsets of X. Finally, when X lies in a real topological vector space, 5(S[A] = %[X] n S[A]. In topologizing hyperspaces (i.e., spaces of subsets), we will use the uniform topology, regarding which we adopt Michael [1] as standard reference. Let X be a uniform space, and let [Ea C A X A|a G ffi) be a fundamental system of symmetric entourages of X. The uniform topology for [X] is the topology generated by declaring &a;[A] = {B E [X]\B C Ea{A) and A C Ea{B)} for each a £ fl to be a nbd of A {A E [X]). By the uniform topology on a hyperspace %[X] C [X] is meant the relative topology of %[X] when [X] carries the uniform topology. 1.0 Definition [2]. Let {S, ffi) be a commutative semigroup and 1ir: R+X S -> S a map such that, denoting ^(A,i) = Xs, Received by the editors December 20, 1974 and, in revised form, August 6, 1975. AMS (MOS) subject classifications (1970). Primary 54B20, 54C25; Secondary 57A17.