Hyperspaces of finite subsets, homeomorphic to pre-Hilbert spaces

Let Fin(X)Fin(X) be the hyperspace consisting of non-empty finite subsets of a topological space X endowed with the Vietoris topology. In this paper, we characterize topological spaces X   whose hyperspace Fin(X)Fin(X) is homeomorphic to the pre-Hilbert space ℓ2f(κ) for an uncountable cardinal κ.