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Best approximation properties in spaces of measurable functions
Author(s) -
Ciesielski Maciej,
Lewicki Grzegorz
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600485
Subject(s) - mathematics , hausdorff space , compact space , banach space , metric space , uniform continuity , measure (data warehouse) , subspace topology , hausdorff distance , hausdorff measure , interpolation space , measurable function , fréchet space , function space , pure mathematics , compact open topology , sequence (biology) , modes of convergence (annotated index) , lorentz space , birnbaum–orlicz space , lorentz transformation , mathematical analysis , topological vector space , topological space , hausdorff dimension , functional analysis , computer science , database , chemistry , genetics , biology , biochemistry , classical mechanics , bounded function , physics , gene , isolated point
We research proximinality of μ‐sequentially compact sets and μ‐compact sets in measurable function spaces. Next we show a correspondence between the Kadec–Klee property for convergence in measure and μ‐compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec–Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one‐dimensional subspace in sequence Lorentz spaces d ( w , 1 ) .