#### Vol. 11, No. 5, 2018

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Time stopping for Tsirelson's norm

### Kevin Beanland, Noah Duncan and Michael Holt

Vol. 11 (2018), No. 5, 857–866
##### Abstract

Tsirelson’s norm $\parallel \cdot {\parallel }_{T}$ on ${c}_{00}$ is defined as the limit of an increasing sequence of norms ${\left(\parallel \cdot {\parallel }_{n}\right)}_{n=1}^{\infty }$. For each $n\in ℕ$ let $j\left(n\right)$ be the smallest integer satisfying $\parallel x{\parallel }_{j\left(n\right)}=\parallel x{\parallel }_{T}$ for all $x$ with $maxsuppx=n$. We show that $j\left(n\right)$ is $O\left({n}^{1∕2}\right)$. This is an improvement of the upper bound of $O\left(n\right)$ given by P. Casazza and T. Shura in their 1989 monograph on Tsirelson’s space.

##### Keywords
Tsirelson's space, Banach space
Primary: 46B03