Hyperspaces of Banach Spaces with the Attouch—Wets Topology |
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Authors: | Katsuro Sakai and Masato Yaguchi |
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Institution: | (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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