Time stopping for Tsirelson's norm

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 $n\in \mathbb{N}$ 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.