Tsirelson space | Wikipedia audio article

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00:01:39 1 Tsirelson's construction
00:04:42 2 Properties
00:06:22 3 Derived spaces
00:06:53 4 See also



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SUMMARY
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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.