We introduce and study a natural class of variable exponent ℓp spaces, which generalizes the classical spaces ℓp and c0. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.
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By Fin(X) (resp. Fink (X)), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ2 (τ) be the Hilbert space with weight τ and ℓf2 (τ) the linear span of the canonical orthonormal basis of ℓ2 (τ). It is shown that if E = ℓf2 (τ) or E is an absorbing set in ℓ2 (τ) for one of the absolute Borel classes aα (τ) and Mα (τ) of weight ≤ τ (α > 0) then Fin(E) and each Fink (E) are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic to E and each Fink (X) is a connected E-manifold.
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