- Tsirelson space
In
mathematics , Tsirelson space "T" is an example of a reflexiveBanach space in which neither an "l p" space nor a "c"0 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.Construction
Let P_n(x) denote the
operator which sets all coordinates x_i, ileq n to zero.We call a sequence x_n}_{n=1}^N "block-disjoint", if for each n there are natural numbers a_n and b_n, so that x_n)_i=0 when i
or i>b_n. Also, a_1leq b_1 . Define these four properties for a set A:
# A is contained in theunit ball of c_0. Everyunit vector e_i is in A.
# forall xin A forall yin c_0 (|y|leq|x|Rightarrow yin A) (pointwise comparison)
# For any N, let x_1,dots,x_N) be a block-disjoint sequence in A, then 1over2}P_N(x_1+cdots+x_N)in A.
# forall xin A exists n 2P_n(x)in A.We define T as the space with
unit ball V, where V is an absolutely convex weaklycompact set, for which (1)-(4) hold true. It may be noted that a set with the given properties exists, but is not unique.Properties
The Tsirelson space is reflexive 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 reflexiveBanach space .References
* B. S. Tsirelson (1974): "Not every Banach space contains an imbedding of l_p or c_0." "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).
* [http://www.tau.ac.il/~tsirel/Research/myspace/remins.html Boris Tsirelson's reminiscences on his web page]
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