Tsirelson space

In mathematics, Tsirelson space "T" is an example of a reflexive Banach space in which neither an "l ^p" space nor a "c"₀ space can be embedded.

It was introduced by B. S. Tsirelson in 1974. In the same year, Figiel and Johnson published a related article; there they used $T$ for the dual of the Tsirelson space.

Construction

Let $P\_n(x)$ denote the operator which sets all coordinates $x\_i$, $ileq\; n$ to zero.

We call a sequence $\{x\_n\}\_\{n=1\}^N$ "block-disjoint", if for each $n$ there are natural numbers $a\_n$ and $b\_n$, so that $(x\_n)\_i=0$ when

Define these four properties for a set $A$:
# $A$ is contained in the unit ball of $c\_0$. Every unit vector $e\_i$ is in $A$.
# $forall\; xin\; A\; forall\; yin\; c\_0\; (|y|leq|x|Rightarrow\; yin\; A)$ (pointwise comparison)
# For any $N$, let $(x\_1,dots,x\_N)$ be a block-disjoint sequence in $A$, then $\{1over2\}P\_N(x\_1+cdots+x\_N)in\; A$.
# $forall\; xin\; A\; exists\; n\; 2P\_n(x)in\; A$.

We define $T$ as the space with unit ball $V$, where $V$ is an absolutely convex weakly compact set, for which (1)-(4) hold true. It may be noted that a set with the given properties exists, but is not unique.

Properties

The Tsirelson space is reflexive and finitely universal. Also, every infinite-dimensional subspace is finitely universal.

Derived spaces

The symmetric Tsirelson space $S(T)$ is polynomially reflexive and it has the approximation property. As with $T$, it is reflexive and no $l^p$ space can be embedded into it.

Since it is symmetric, it can be defined even on an uncountable supporting set, giving an example of non-separable polynomially reflexive Banach space.

References

* B. S. Tsirelson (1974): "Not every Banach space contains an imbedding of $l\_p$ or $c\_0$." "Functional Anal. Appl." 8(1974), 138–141
* T. Figiel, W. B. Johnson (1974): "A uniformly convex Banach space which contains no $l\_p$." "Composito Math." 29(1974).
* V. Spinka (2002): "Smoothness on Banach spaces". Diploma work, Charles University Prague, Department of Mathematical Analysis. (Proof of the polynomial reflexivity of $S(T)$ for both separable and non-separable cases).
* [http://www.tau.ac.il/~tsirel/Research/myspace/remins.html Boris Tsirelson's reminiscences on his web page]