English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A result of G. Godefroy asserts that a Banach space $X$ contains anisomorphic copy of $\ell_1$ if and only if there is an equivalent norm$|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ andevery $\varepsilon>0$, there exists $x\in S_X$ so that $|||y+r x|||\geq(1-\varepsilon)(|||y|||+\vert r\vert)$ for every $y\in Y$ and every$r\in\mathbb R$. In this paper we generalize this result to larger cardinals,showing that if $\kappa$ is an uncountable cardinal then a Banach space $X$contains a copy of $\ell_1(\kappa)$ if and only if there is an equivalent norm$|||\cdot|||$ on $X$ such that for every subspace $Y$ of $X$ with$dens(Y)<\kappa$ there exists a norm-one vector $x$ so that $||| y+rx|||=|||y|||+\vert r\vert$ whenever $y\in Y$ and $r\in\mathbb{R}$. This resultanswers a question posed by S. Ciaci, J. Langemets and A. Lissitsin, where theauthors wonder whether the previous statement holds for infinite succesorcardinals. We also show that, in the countable case, the result of Godefroycannot be improved to take $\varepsilon=0$.

A renorming characterization of Banach spaces containing $\ell_1(\kappa)$