Цирельсон кеңістігі: функционалдық анализдегі жаңа үлгі

In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is reflexive. It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article 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 ℓ p space, 1 ≤ p < ∞, or to c0. All classical Banach spaces known to , 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 ℓ p 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 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 , 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.

The Tsirelson space T* is reflexive 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 1 + ε for every ε > 0. 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 Tsirelson space T is distortable, but it is not known whether it is arbitrarily distortable. 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 ℓ p and c0. The dual space T is not minimal. The space T* is polynomially reflexive.

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.