Extension operators on balls and on spaces of finite sets

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We study extension operators between spaces σₙ(2ˣ) of subsets of X of cardinality at most n. As an application, we show that if Bₕ is the unit ball of a nonseparable Hilbert space H, equipped with the weak topology, then, for any 0<λ<μ, there is no extension operator T: C(λBₕ)→ C(μBₕ).