Embedding Theorems for Classes of Convex Sets in a Hypernormed Vector Space

Abstract

As a partial extension of Rädström's einbedding theorem, Fischer recently proved that the class of all nonempty, hypernorm sequentially compact, convex subsets of a hypernormed vector space can be embedded as a convex cone in a vector space, and he also obtained a similar embedding theorem for the class of all hypernorm balls. In the present paper, Fischer's results are improved by taking into account the inclusion of sets as an order relation, and it is also shown that the füll Statement of Rädström's embedding theorem cannot be extended to the general hypernormed case. Furthermore, embedding theorems are proven for the class of all order intervals of a vector lattice. These results are roore precise than Fischer's embedding theorem for hypernorm balls and cannot be extended to the general case. AMS 1980 subject Classification: 52A05, 52A07, 06F05, 06F20, 46A40, 46B30. Brought to you by | Michigan State University Authenticated Download Date | 6/26/15 8:20 PM 58 Schmidt 1. INTRODUCTIONIn a recent paper published in this journal, Fischer [2] proved that the class of all nonempty, hypernorm sequentially compact, convex subsets of a hypernormed vector space can be embedded as a convex cone in a suitable vector space. Under the assumption that the hypernorm is splittable, he also obtained a similar embedding theorem for the class of all hypernorm balls. While the first of these results is a partial extension of Rädström' embedding theorem [12] for the class of all nonempty, compact, convex subsets of anormed vector space, the second one contains embedding theorems for the class of all norm balls of a normed vector space and for the class of all order intervals of a vector lattice. These special cases are of particular interest in interval mathematics [1,5,6,8,9,10,11].The usefulness of such embedding theorems clearly depends on the amount of Information they provide on the structure of the embedding vector space and the properties of the embedding map. While Fischer's results only guarantee the existence of an embedding vector space and an embedding map which is additive and positively homogeneous. Rädström's embedding theorem and its generalizations also establish the preservation of topological properties. Furthermore, Kaucher [5] proved abstract embedding theorems which ensure the preservation of order properties, and embedding theorems for classes of convex sets in a Hausdorff locally convex vector Space which simultaneously guarantee the preservation of topological and order properties were proven by Hörmander [4] and Schmidt [14]. In view of these remarks, it is a natural question to ask whether or not Fischer's results can be improved. This question will be studied in the present paper. Brought to you by | Michigan State University Authenticated Download Date | 6/26/15 8:20 PM Embedding theorems 59This paper is organized as follows:In Sections 2 and 3, we recall some definitions concerning cones and hypernormed vector spaces.In the first part of Section 4, we study the question of whether or not the füll Stat...