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Embedding Banach spaces into the space of bounded functions with countable support
Author(s) -
Johnson William B.,
Kania Tomasz
Publication year - 2019
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.201800308
Subject(s) - mathematics , countable set , separable space , banach space , bounded function , subspace topology , embedding , space (punctuation) , pure mathematics , basis (linear algebra) , discrete mathematics , set (abstract data type) , mathematical analysis , computer science , geometry , artificial intelligence , programming language , operating system
We prove that a WLD subspace of the spaceℓ ∞ c ( Γ )consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into ℓ ∞ if and only if it does not contain isometric copies ofc 0 ( ω 1 ) . Moreover, a subspace ofℓ ∞ c ( ω 1 )is constructed that has an unconditional basis, does not embed into ℓ ∞ , and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies ofc 0 ( ω 1 ) ).