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  • 1
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    The Szlenk index of Lp(X) (2014)
    Oxford University Press
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    Publication Date: 2014-03-16
    Description: We find an optimal upper bound on the values of the weak $^* $ -dentability index $ {{\mathrm {Dz}}} (X)$ in terms of the Szlenk index $ {{\mathrm {Sz}}} (X)$ of a Banach space $X$ with separable dual. Namely, if $ {{\mathrm {Sz}}} (X) = \omega ^{\alpha }$ , for some $\alpha 〈 \omega _1,$ and $p\in (1,\infty )$ , then \[ {{\mathrm {Sz}}} (X)\le {{\mathrm {Dz}}} (X)\le {{\mathrm {Sz}}} (L_p(X))\le \left\{ \begin {array}{ll}\omega ^{\alpha +1} & {\mathrm {if\ }} {\alpha }{\mathrm {\ is\ a\ finite\ ordinal,}}\\ \omega ^{\alpha } & {\mathrm {if\ }} {\alpha }{\mathrm {\ is\ an\ infinite\ ordinal.}} \end {array}\right .\]
    Print ISSN: 0024-6093
    Electronic ISSN: 1469-2120
    Topics: Mathematics
    Published by Oxford University Press
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  • 2
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    On coarse embeddings into c0(Γ) (2018)
    Oxford University Press
    Details
    Publication Date: 2018-03-06
    Description: Let λ be a large enough cardinal number (assuming the Generalized Continuum Hypothesis it suffices to let λ=ℵω). If X is a Banach space with dens(X)≥λ, which admits a coarse (or uniform) embedding into any c0(Γ), then X fails to have non-trivial cotype, i.e. X contains ℓ∞n C -uniformly for every C〉1. In the special case when X has a symmetric basis, we may even conclude that it is linearly isomorphic with c0(densX).
    Print ISSN: 0033-5606
    Electronic ISSN: 1464-3847
    Topics: Mathematics
    Published by Oxford University Press
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