# Distorting Banach spaces

• Professor Kevin Beanland (Washington & Lee University, Virginia, USA)
• Wednesday 08 March 2017, 14:00-15:00
• CMS, MR14.

A Banach space $X$ with a norm $\|\cdot\|$ is called D-distortable if there is an equivalent norm $|\cdot|$ on $X$ so for each infinite-dimensional subspace $Y$ of $X$ there are vectors $x,y \in Y$ with $\|x\|=\|y\|=1$ and $|x|/|y|>D$. A space is arbitrarily distortable if it is D-distortable for every $D>1$. A result of R.C. James from the 1960s shows that the Banach spaces $\ell_1$ and $c_0$ are not distortable for any $D>1$. Shortly after this V. Milman showed that if a Banach space does not contain any $\ell_p$ or $c_0$ it must have a subspace that is $D$-distortable for some $D>1$. In the 1990s it was shown explicitly by Odell and Schlumprecht that Tsirelson’s famous space was itself $D$-distortable for each \$D

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