For studying strong uniqueness of best simultaneous approximants, we consider three properties of the triplets (X, V, F) where X is a normed linear space, V is a nonempty closed subset of X and 7 is a family of nonempty closed and bounded subsets of X. These properties are called SUBSA, α-SUBSA and SUBSA of order q. Here we explore examples satisfying these properties. The classical Haar theory for C_0(T) as well as the more recent Haar theory for the function space C_0(T, H) of Hilbert space-valued continuous functions are extended to best simultaneous approximants of certain sets.
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