Hyperspaces of Banach Spaces with the Attouch–Wets Topology

摘要

Let X be an infinite-dimensional Banach space with weight τ. By Cld_(AW)(X), we denote the hyperspace of nonempty closed sets in X with the Attouch–Wets topology. By Fin_(AW)(X), Comp_(AW)(X) and Bdd_(AW)(X), we denote the subspaces of CldAW(X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved that Fin_(AW)(X)≈Comp_(AW)(X)≈l_2(τ)*l_2~f and Bdd_(AW)(X)≈l_2(2~τ)*l_2~f, where ≈ means ‘is homeomorphic to’, l_2(τ) is the Hilbert space with weight τ (l_2(N_0)=l_2 the separable Hilbert space) and l_2~f={(x_i)_(i∈N)∈l_2|x_i=0 except for finitely many i∈N}.