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 Jonhson published a related article; there they used T for the dual of the Tsirelson space.

Construction

Let P_n(x) denote the operator, which sets all coordinates x_i, to zero.

We call a sequence block-disjoint, if for each n there are natural numbers a_n and b_n, so that (x_n)_i = 0 when i < a_n or i > b_n. Also, .

Define these four properties for a set A:

1. A is contained in the unit ball. Every unit vector e_i is in A.
2. (pointwise)
3. For any N, let be a block-disjoint sequence in A, then .
4. .

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.

Properties

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

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 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.

References

• B. S. Tsirelson (1974): Not every Banach space contains an imbedding of l_p or c₀. Functional Anal. Appl. 8(1974), 138–141
• T. Figiel, W. B. Johnson (1974): A uniformly convex Banach space which contains no l_p. 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).