Tsirelson space

Example

Let $c_{00}$ be the space of real sequences with finitely many nonzero terms. For finite sets $E, F \subset \N$, write $E < F$ when $\max E < \min F$. For $x \in c_{00}$ and a finite set $E$, let $E x$ be the sequence equal to $x$ on $E$ and to $0$ off $E$. Call finite sets $E_1, \dots, E_n$ admissible when $n \le \min E_1$ and $E_1 < E_2 < \cdots < E_n$. There is exactly one norm on $c_{00}$ that satisfies $$\norm{x} = \max \Big( \norm{x}_\infty, \ \frac{1}{2} \sup \sum_{j=1}^n \norm{E_j x} \Big),$$ where the supremum runs over all $n$ and all admissible $E_1, \dots, E_n$. The completion $T$ of $c_{00}$ for this norm is the Tsirelson space. It is a Banach space.

Check of two values. For a unit vector $e_k$, at most one set $E_j$ meets $k$, so the sum is at most $\norm{e_k}$, and the equation gives $\norm{e_k} = 1$. For $x = e_{n+1} + \cdots + e_{2n}$, the sets $\set{n+1}, \dots, \set{2n}$ are admissible, so $n/2 \le \norm{x} \le n$.