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arXiv · 1604.04131

Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals

Marek Cúth·Michal Johanis
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Abstract

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.

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Marek Cúth, Michal Johanis. 2016-04-14. Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals. https://doi.org/10.1090/proc/13590

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