We solve a problem of Lacey (1973) by showing that there exist a metrizable compact space K and a closed space H ⊂ C(K) containing constants with ∂ H K = K such that H is maximal with respect to ∂ H K and H is not a Lindenstrauss space.
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We give a sufficient condition for the set of nowhere monotone measures to be a residual G δ set in a subspace of signed Radon measures on a locally compact second-countable Hausdorff space with no isolated points. We prove that the set of nowhere monotone signed Radon measures on a ddimensional real space R d is lineable. More specifically, we prove that there exists a vector space of dimension c (the cardinality of the continuum) of signed Radon measures on R d every non-zero element of which is a nowhere monotone measure that is almost everywhere differentiable with respect to the d-dimensional Lebesgue measure. Using this result we show that the set of these measures is even maximal dense-lineable in the space of bounded signed Radon measures on R d that are almost everywhere differentiable with respect to the d-dimensional Lebesgue measure.
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