On Extending Mappings into Nonlocally Convex Linear Metric Spaces

Abstract

Abstract.It is proved that the following spaces are absolute retracts: every F-space with a Schauder basis and certain function spaces along with their subgroups of integer-valued elements. It is also observed that for every o-compact convex set, the absolute extension property for compacta implies the AR-property.1. Introduction. The purpose of this paper is to provide new examples of infinite-dimensional ANRs. Detecting the ANR-property of convex subsets of nonlocally convex metric linear spaces and topological groups is of great importance. For example, the topological classification of these spaces, given recently in [4, 5 and 3], required the ANR-property. We prove that the following spaces are absolute retracts: (1) every complete metric linear space (= F-space) with a Schauder basis, (2) certain function spaces which include Lp (p 5= 0) and Orlicz spaces, and (3) additive subgroups consisting of all integer-valued functions in certain function spaces. Consequently, each of these spaces, when complete and separable, is homeomorphic to a Hubert space [4, 5]. The argument used in verifying the AR-property of the above examples is also employed to show that the AR and the AE(fé') (absolute extension property for compacta) properties coincide for a-compact convex sets. This enables us to find a dense convex topological copy of 2, the linear span of the Hilbert cube in the Hilbert space l2, in every separable infinite-dimensional complete convex set.Our approach is very elementary and mostly involves the natural equiconnected structures of convex sets and contractible groups. We also employ the admissibility

Continuous Selections of Multivalued Mappings

“…To emphasize the difference on a more nontrivial level, let us recall that for a wide class of nonlocally convex, completely metrizable, topological vector spaces it was proved in [26] that all such spaces are absolute retracts (with respect to all metrizable spaces), abbreviated as AR ′ s. At the same time, at present there is no known example of a nonlocally convex, completely metrizable, topological vector space E which can be successfully substituted instead of Banach (or Fréchet) spaces B into the assumption of the Michael selection Theorem 1. In particular, Dobrowolski stated (private communication) the following: Problem 3.…”