Hyperspaces of Banach Spaces with the Attouch—Wets Topology |
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Authors: | Katsuro Sakai and Masato Yaguchi |
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Affiliation: | (1) Institute of Mathematics, University of Tsukuba, 305-8571 Tsukuba-City, Japan |
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Abstract: | 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 Cld AW (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)≈ℓ2(τ)×ℓ2 f ℓandℓBdd AW (X)≈ℓ2(2τ)×ℓ2 f where ≈ means ‘is homeomorphic to’, ℓ2(τ) is the Hilbert space with weight τ (ℓ2(ℵ0)=ℓ2 the separable Hilbert space) and ℓ2 f ={(x i ) iεN εℓ2∣x i =0 except for finitely many iεN}. |
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Keywords: | The Attouch— Wets topology the hyperspace of finite sets the hyperspace of compact sets the hyperspace of bounded closed sets Banach space ℓ 2(τ )×ℓ 2 f |
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