Tsirelson Space
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the Tsirelson space is the first example of a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
in which neither an  ''p'' space nor a ''c''0 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 () where they used the notation ''T'' for the ''dual'' of Tsirelson's example. Today, the letter ''T'' is the standard notationsee for example , p. 8; , p. 95; ''The Handbook of the Geometry of Banach Spaces'', vol. 1, p. 276; vol. 2, p. 1060, 1649. 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
, as Banach space, to an ''ℓ'' ''p'' space, 1 ≤ ''p'' < ∞, or to ''c''0. All classical Banach spaces known to , spaces of
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s, of differentiable functions or of integrable functions, and all the Banach spaces used in functional analysis for the next forty years, contain some ''ℓ'' ''p'' or ''c''0. 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 ''ℓ'' ''p'' or to ''c''0. Moreover, it was shown by Baudier, Lancien, and Schlumprecht that ''ℓ'' ''p'' and ''c''0 do not even coarsely embed into T*. The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Thomas Schlumprecht (), 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.


Tsirelson's construction

On the vector space ℓ of bounded scalar sequences , let ''P''''n'' denote the
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
which sets to zero all coordinates ''x''''j'' of ''x'' for which ''j'' ≤ ''n''. A finite sequence \_^N of vectors in ℓ is called ''block-disjoint'' if there are natural numbers \textstyle \_^N so that a_1 \leq b_1 < a_2 \leq b_2 < \cdots \leq b_N, and so that (x_n)_i=0 when i or 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 In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
). The crucial step in the Tsirelson construction is to let ''K'' be the ''smallest'' pointwise closed subset of  ''B''  satisfying the following two properties:conditions b, c, d here are conditions (3), (2) and (4) respectively in , and a is a modified form of condition (1) from the same article. :a. For every integer  ''j''  in N, the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
''e''''j'' and all multiples \lambda e_j, for , λ,  ≤ 1, belong to ''K''. :b. For any integer ''N'' ≥ 1, if \textstyle (x_1,\ldots,x_N) is a block-disjoint sequence in ''K'', then \textstyle belongs to ''K''. This set ''K'' satisfies the following stability property: :c. Together with every element ''x'' of ''K'', the set ''K'' contains all vectors ''y'' in ℓ such that , ''y'',  ≤ , ''x'', (for the pointwise comparison). It is then shown that ''K'' is actually a subset of ''c''0, the Banach subspace of ℓ consisting of scalar sequences tending to zero at infinity. This is done by proving that :d: for every element ''x'' in ''K'', there exists an integer ''n'' such that 2 ''P''''n''(''x'') belongs to ''K'', and iterating this fact. Since ''K'' is pointwise compact and contained in ''c''0, it is weakly compact in ''c''0. Let ''V'' be the closed convex hull of ''K'' in ''c''0. It is also a weakly compact set in ''c''0. It is shown that ''V'' satisfies b, c and d. The Tsirelson space ''T''* is the Banach space whose unit ball is ''V''. The unit vector basis is an unconditional basis for ''T''* and ''T''* is reflexive. Therefore, ''T''* does not contain an isomorphic copy of ''c''0. The other ''ℓ'' ''p'' spaces, 1 ≤ ''p'' < ∞, are ruled out by condition b.


Properties

The Tsirelson space is reflexive () and finitely universal, which means that for some constant , the space contains -isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space , there exists a subspace of the Tsirelson space with multiplicative Banach–Mazur distance to less than . Actually, every finitely universal Banach space contains ''almost-isometric'' copies of every finite-dimensional normed space, meaning that can be replaced by for every . Also, every infinite-dimensional subspace of is finitely universal. On the other hand, every infinite-dimensional subspace in the dual of contains almost isometric copies of \scriptstyle, the -dimensional ℓ1-space, for all . The Tsirelson space is distortable, but it is not known whether it is arbitrarily distortable. The space is a ''minimal'' Banach space. This means that every infinite-dimensional Banach subspace of contains a further subspace isomorphic to . Prior to the construction of , the only known examples of minimal spaces were ''ℓ'' ''p'' and 0. The dual space is not minimal.see , p. 56. The space is polynomially reflexive.


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 In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
.


See also

* Distortion problem *
Sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
,
Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
*
James' space In the area of mathematics known as functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner pr ...


Notes


References

* . * * . * . * . * . * . * . * . * . English translation in Russian Math. Surveys 25 (1970), 111-170. * . * .


External links


Boris Tsirelson's reminiscences on his web page
{{Functional analysis Banach spaces