The same year, Figiel and Johnson published a related article (Figiel & Johnson (1974)) where they used the notation T for the dual of Tsirelson's example.
Today, the letter T is the standard notation[1] for the dual of the original example, while the original Tsirelson example is denoted by T*.
Also, new attempts in the early '70s[2] to promote a geometric theory of Banach spaces led to ask [3] whether or not every infinite-dimensional Banach space has a subspace isomorphic to some ℓ p or to c0.
Moreover, it was shown by Baudier, Lancien, and Schlumprecht that ℓ p and c0 do not even coarsely embed into T*.
The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Thomas Schlumprecht (Schlumprecht (1991)), on which depend Gowers' solution to Banach's hyperplane problem[4] and the Odell–Schlumprecht solution to the distortion problem.
Also, several results of Argyros et al.[5] are based on ordinal refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.
[6] On the vector space ℓ∞ of bounded scalar sequences x = {xj } j∈N, let Pn denote the linear operator which sets to zero all coordinates xj of x for which j ≤ n. A finite sequence
of vectors in ℓ∞ is called block-disjoint if there are natural numbers
The crucial step in the Tsirelson construction is to let K be the smallest pointwise closed subset of B∞ satisfying the following two properties:[7] This set K satisfies the following stability property: It is then shown that K is actually a subset of c0, the Banach subspace of ℓ∞ consisting of scalar sequences tending to zero at infinity.
It is shown that V satisfies b, c and d. The Tsirelson space T* is the Banach space whose unit ball is V. The unit vector basis is an unconditional basis for T* and T* is reflexive.
The other ℓ p spaces, 1 ≤ p < ∞, are ruled out by condition b.
The Tsirelson space T* is reflexive (Tsirel'son (1974)) and finitely universal, which means that for some constant C ≥ 1, the space T* contains C-isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space X, there exists a subspace Y of the Tsirelson space with multiplicative Banach–Mazur distance to X less than C. Actually, every finitely universal Banach space contains almost-isometric copies of every finite-dimensional normed space,[8] meaning that C can be replaced by 1 + ε for every ε > 0.
On the other hand, every infinite-dimensional subspace in the dual T of T* contains almost isometric copies of
Prior to the construction of T*, the only known examples of minimal spaces were ℓ p and c0.
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.