Extension operators on balls and on spaces of finite sets

We study extension operators between spaces $σ_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ 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 $T: C(λB_H)\to C(μB_H)$.

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