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It was introduced by B. S. Tsirelson in 1974. 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 for the dual of the original example, while the original Tsirelson example is denoted by T*. In T* or in T, no subspace is isomorphic, as Banach space, to an lp space, 1 ≤ p < ?, or to c0.
All classical Banach spaces known to Banach (1932), spaces of continuous functions, of differentiable functions or of integrable functions, and all the Banach spaces used in functional analysis for the next forty years, contain some lp or c0. Also, new attempts in the early '70s to promote a geometric theory of Banach spaces led to ask  whether or not every infinite-dimensional Banach space has a subspace isomorphic to some lp or to c0.
The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Schlumprecht (Schlumprecht (1991)), on which depend Gowers' solution to Banach's hyperplane problem and the Odell–Schlumprecht solution to the distortion problem. Also, several results of Argyros et al. are based on ordinal refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.
On the vector space l? of bounded scalar sequences , 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 so that , and so that when or , for each n from 1 to N.
a. For every integer j in N, the unit vectorej and all multiples , for |λ| ≤ 1, belong to K.
b. For any integer N ≥ 1, if is a block-disjoint sequence in K, then belongs to K.
This set K satisfies the following stability property:
c. Together with every element x of K, the set K contains all vectors y in l? such that |y| ≤ |x| (for the pointwise comparison).
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. This is done by proving that
d: for every element x in K, there exists an integer n such that 2 Pn(x) belongs to K,
and iterating this fact. Since K is pointwise compact and contained in c0, it is weakly compact in c0. Let V be the closed convex hull of K in c0. It is also a weakly compact set in c0. 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. Therefore, T* does not contain an isomorphic copy of c0. The other lp 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 , 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, meaning that C can be replaced by for every . Also, every infinite-dimensional subspace of T* is finitely universal. On the other hand, every infinite-dimensional subspace in the dual T of T* contains almost isometric copies of , the n-dimensional l1-space, for all n.
The space T* is a minimal Banach space. This means that every infinite-dimensional Banach subspace of T* contains a further subspace isomorphic to T*. Prior to the construction of T*, the only known examples of minimal spaces were lp and c0. The dual space T is not minimal.
^The question is whether every infinite-dimensional Banach space is isomorphic to its hyperplanes. The negative solution is in Gowers, "A solution to Banach's hyperplane problem". Bull. London Math. Soc. 26 (1994), 523-530.
^for example, S. Argyros and V. Felouzis, "Interpolating Hereditarily Indecomposable Banach spaces", Journal Amer. Math. Soc., 13 (2000), 243-294; S. Argyros and A. Tolias, "Methods in the theory of hereditarily indecomposable Banach spaces", Mem. Amer. Math. Soc. 170 (2004), no. 806.
^S. Argyros and R. Haydon constructed a Banach space on which every bounded operator is a compact perturbation of a scalar multiple of the identity, in "A hereditarily indecomposable L?-space that solves the scalar-plus-compact problem", Acta Mathematica (2011) 206: 1-54.
^conditions b, c, d here are conditions (3), (2) and (4) respectively in Tsirel'son (1974), and a is a modified form of condition (1) from the same article.
^this is because for every n, C and ε, there exists N such that every C-isomorph of l?N contains a -isomorph of l?n, by James' blocking technique (see Lemma 2.2 in Robert C. James "Uniformly Non-Square Banach Spaces", Annals of Mathematics, Vol. 80, 1964, pp. 542-550), and because every finite-dimensional normed space -embeds in l?n when n is large enough.
Lindenstrauss, Joram (1971), "The geometric theory of the classical Banach spaces", Actes du Congrès Intern. Math., Nice 1970: 365-372.
Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Berlin: Springer-Verlag, ISBN3-540-08072-4.
Milman, V. D. (1970), "Geometric theory of Banach spaces. I. Theory of basic and minimal systems", Uspekhi Mat. Nauk (in Russian), 25 no. 3: 113-174. English translation in Russian Math. Surveys 25 (1970), 111-170.