Tsirelson space

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In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l p space nor a c0 space can be embedded.

It was introduced by B. S. Tsirelson in 1974. In the same year, Figiel and Johnson published a related article; there they used T for the dual of the Tsirelson space.

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[edit] Construction

Let Pn(x) denote the operator which sets all coordinates xi, i\leq n to zero.

We call a sequence \{x_n\}_{n=1}^N block-disjoint, if for each n there are natural numbers an and bn, so that (xn)i = 0 when i < an or i > bn. Also, a_1\leq b_1<a_2\leq b_2<\cdots.

Define these four properties for a set A:

  1. A is contained in the unit ball. Every unit vector ei is in A.
  2. \forall x\in A\ \forall y\in m\ (|y|\leq|x|\Rightarrow y\in A) (pointwise)
  3. For any N, let (x_1,\dots,x_N) be a block-disjoint sequence in A, then {1\over2}P_N(x_1+\cdots+x_N)\in A.
  4. \forall x\in A\ \exists n\ 2P_n(x)\in A.

We define T as the space with unit ball V, where V is an absolutely convex weak compact set, for which (1)-(4) hold true.

[edit] Properties

The Tsirelson space is reflexive, it is uniformly convex and finitely universal. Also, every infinite-dimensional subspace is finitely universal.

[edit] Derived spaces

The symmetric Tsirelson space S(T) is polynomially reflexive and it has the approximation property. As with T, it is reflexive and no lp 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.

[edit] References

  • B. S. Tsirelson (1974): Not every Banach space contains an imbedding of lp or c0. Functional Anal. Appl. 8(1974), 138–141
  • T. Figiel, W. B. Johnson (1974): A uniformly convex Banach space which contains no lp. Composito Math. 29(1974).
  • V. Spinka (2002): Smoothness on Banach spaces. Diploma work, Charles University Prague, Department of Mathematical Analysis. (Proof of the polynomial reflexivity of S(T) for both separable and non-separable cases).
  • Boris Tsirelson's web page
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