# Tsirelson space

Last updated

In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an  p space nor a c0 space can be embedded. The Tsirelson space is reflexive.

## Contents

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 [1] 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  p 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  p or c0. Also, new attempts in the early '70s [2] to promote a geometric theory of Banach spaces led to ask [3] whether or not every infinite-dimensional Banach space has a subspace isomorphic to some  p 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 [4] and the OdellSchlumprecht solution to the distortion problem. Also, several results of Argyros et al. [5] are based on ordinal refinements of the Tsirelson construction, culminating with the solution by ArgyrosHaydon of the scalar plus compact problem. [6]

## Tsirelson's construction

On the vector space ℓ of bounded scalar sequences x ={xj}jN, let Pn denote the linear operator which sets to zero all coordinates xj of x for which j  n.

A finite sequence ${\displaystyle \{x_{n}\}_{n=1}^{N}}$ of vectors in ℓ is called block-disjoint if there are natural numbers ${\displaystyle \textstyle \{a_{n},b_{n}\}_{n=1}^{N}}$ so that ${\displaystyle a_{1}\leq b_{1}, and so that ${\displaystyle (x_{n})_{i}=0}$ when ${\displaystyle i or ${\displaystyle i>b_{n}}$, 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). The crucial step in the Tsirelson construction is to let K be the smallest pointwise closed subset of B satisfying the following two properties: [7]

a. For every integer j in N, the unit vector ej and all multiples ${\displaystyle \lambda e_{j}}$, for |λ|  1, belong to K.
b. For any integer N  1, if ${\displaystyle \textstyle (x_{1},\ldots ,x_{N})}$ is a block-disjoint sequence in K, then ${\displaystyle \textstyle {{1 \over 2}P_{N}(x_{1}+\cdots +x_{N})}}$ 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 c0, 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 2Pn(x) belongs to K,

and iterating this fact. Since K is pointwise compact and contained in c0, it is weakly compact in c0. Let V be the closed convex hull of K in c0. It is also a weakly compact set in c0. 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 c0. The other  p spaces, 1  p < ∞, are ruled out by condition b.

## Properties

The Tsirelson space T* is reflexive (Tsirel'son (1974)) and finitely universal, which means that for some constant C 1, the space T* contains C-isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space X, there exists a subspace Y of the Tsirelson space with multiplicative BanachMazur distance to X less than C. Actually, every finitely universal Banach space contains almost-isometric copies of every finite-dimensional normed space, [8] meaning that C can be replaced by 1 + ε for every ε > 0. Also, every infinite-dimensional subspace of T* is finitely universal. On the other hand, every infinite-dimensional subspace in the dual T of T* contains almost isometric copies of ${\displaystyle \scriptstyle {\ell _{n}^{1}}}$, the n-dimensional ℓ1-space, for all n.

The Tsirelson space T is distortable, but it is not known whether it is arbitrarily distortable.

The space T* is a minimal Banach space. [9] This means that every infinite-dimensional Banach subspace of T* contains a further subspace isomorphic to T*. Prior to the construction of T*, the only known examples of minimal spaces were  p and c0. The dual space T is not minimal. [10]

The space T* is polynomially reflexive.

## Derived spaces

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

## Notes

1. see for example Casazza & Shura (1989), p. 8; Lindenstrauss & Tzafriri (1977), p. 95; The Handbook of the Geometry of Banach Spaces, vol. 1, p. 276; vol. 2, p. 1060, 1649.
2. The question is formulated explicitly in Lindenstrauss (1970), Milman (1970), Lindenstrauss (1971) on last page. Lindenstrauss & Tzafriri (1977), p. 95, say that this question was "a long standing open problem going back to Banach's book" (Banach (1932)), but the question does not appear in Banach's book. However, Banach compares the linear dimension of  p to that of other classical spaces, a somewhat similar question.
3. The question is whether every infinite-dimensional Banach space is isomorphic to its hyperplanes. The negative solution is in Gowers, "A solution to Banach's hyperplane problem". Bull. London Math. Soc. 26 (1994), 523-530.
4. for example, S. Argyros and V. Felouzis, "Interpolating Hereditarily Indecomposable Banach spaces", Journal Amer. Math. Soc., 13 (2000), 243–294; S. Argyros and A. Tolias, "Methods in the theory of hereditarily indecomposable Banach spaces", Mem. Amer. Math. Soc. 170 (2004), no. 806.
5. S. Argyros and R. Haydon constructed a Banach space on which every bounded operator is a compact perturbation of a scalar multiple of the identity, in "A hereditarily indecomposable L-space that solves the scalar-plus-compact problem", Acta Mathematica (2011) 206: 1-54.
6. conditions b, c, d here are conditions (3), (2) and (4) respectively in Tsirel'son (1974), and a is a modified form of condition (1) from the same article.
7. this is because for every n, C and ε, there exists N such that every C-isomorph of ℓN contains a (1 + ε)-isomorph of ℓn, by James' blocking technique (see Lemma 2.2 in Robert C. James "Uniformly Non-Square Banach Spaces", Annals of Mathematics, Vol. 80, 1964, pp. 542-550), and because every finite-dimensional normed space (1 + ε)-embeds in ℓn when n is large enough.
8. see Casazza & Shura (1989), p. 54.
9. see Casazza & Shura (1989), p. 56.

## Related Research Articles

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 is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, specifically in functional analysis, a C-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on , together with the vector space structure of pointwise addition and scalar multiplication by constants.

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

1. It is nonnegative, that is for every vector x, one has
2. It is positive on nonzero vectors, that is,
3. For every vector x, and every scalar one has
4. The triangle inequality holds; that is, for every vectors x and y, one has

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual.

In mathematics, an invariant subspace of a linear mapping T : VV from some vector space V to itself, is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.

In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true.

In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space.

In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces.

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 numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

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 makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.

Riesz's lemma is a lemma in functional analysis. It specifies conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when one is not in an inner product space.

In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids.

In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives.

In the area of mathematics known as functional analysis, James' space is an important example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerning the structure of general Banach spaces. The space was first introduced in 1950 in a short paper by Robert C. James.

In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below, under which it forms a Banach space. Orlicz sequence spaces generalize the spaces, and as such play an important role in functional analysis.

## References

• Tsirel'son, B. S. (1974), "'Not every Banach space contains an imbedding of  p or c0", Functional Analysis and Its Applications, 8: 138–141, doi:10.1007/BF01078599, MR   0350378 .
• Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations](PDF). Monografie Matematyczne (in French). 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl   0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
• Figiel, T.; Johnson, W. B. (1974), "A uniformly convex Banach space which contains no  p", Compositio Mathematica, 29: 179–190, MR   0355537 .
• Casazza, Peter G.; Shura, Thaddeus J. (1989), Tsirelson's Space, Lecture Notes in Mathematics, 1363, Berlin: Springer-Verlag, ISBN   3-540-50678-0, MR   0981801 .
• Johnson, William B.; J. Lindenstrauss, Joram, eds. (2001), Handbook of the Geometry of Banach Spaces, 1, Elsevier.
• Johnson, William B.; J. Lindenstrauss, Joram, eds. (2003), Handbook of the Geometry of Banach Spaces, 2, Elsevier.
• Lindenstrauss, Joram (1970), "Some aspects of the theory of Banach spaces", Advances in Mathematics , 5: 159–180, doi:.
• Lindenstrauss, Joram (1971), "The geometric theory of the classical Banach spaces", Actes du Congrès Intern. Math., Nice 1970: 365–372.
• Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Berlin: Springer-Verlag, ISBN   3-540-08072-4 .
• Milman, V. D. (1970), "Geometric theory of Banach spaces. I. Theory of basic and minimal systems", Uspekhi Mat. Nauk (in Russian), 25 no. 3: 113–174. English translation in Russian Math. Surveys 25 (1970), 111-170.
• Schlumprecht, Thomas (1991), "An arbitrary distortable Banach space", Israel Journal of Mathematics , 76: 81–95, arXiv:, doi:, ISSN   0021-2172, MR   1177333 .