Tsirelson space

In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓp space nor a c0 space can be embedded. The Tsirelson space is reflexive. In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓp 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 (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  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 { x n } n = 1 N {displaystyle {x_{n}}_{n=1}^{N}} of vectors in ℓ∞ is called block-disjoint if there are natural numbers { a n , b n } n = 1 N {displaystyle extstyle {a_{n},b_{n}}_{n=1}^{N}} so that a 1 ≤ b 1 < a 2 ≤ b 2 < … ≤ b N {displaystyle a_{1}leq b_{1}<a_{2}leq b_{2}<ldots leq b_{N}} , and so that ( x n ) i = 0 {displaystyle (x_{n})_{i}=0} when i < a n {displaystyle i<a_{n}} or i > b n {displaystyle i>b_{n}} , for each n from 1 to N. The unit ball  B∞  of ℓ∞ is compact and metrizable for the topology of pointwise convergence (the product topology). The crucial step in the Tsirelson construction is to let K be the smallest pointwise closed subset of  B∞  satisfying the following two properties: 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. This is done by proving that