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| In [[mathematics]], especially in [[functional analysis]], the '''Tsirelson space''' is the first example of a [[Banach space]] in which neither an [[lp space|ℓ<sup>''p''</sup> space]] nor a [[lp space|''c''<sub>0</sub> space]] can be embedded. The Tsirelson space is [[reflexive space|reflexive]].
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| It was introduced by [[B. S. Tsirelson]] in 1974. The same year, Figiel and Johnson published a related article ({{harvtxt|Figiel|Johnson|1974}}) where they used the notation ''T'' for the ''dual'' of Tsirelson's example. Today, the letter ''T'' is the standard notation<ref name="Standard">see for example {{harvtxt|Casazza|Shura|1989}}, p. 8; {{harvtxt|Lindenstrauss|Tzafriri|1977}}, p. 95; the Handbook of the Geometry of Banach Spaces, vol. 1, p. 276; vol. 2, p. 1060, 1649.</ref> 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 ℓ<sup>p</sup> space, 1 ≤ ''p'' < ∞, or to ''c''<sub>0</sub>.
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| All classical Banach spaces known to {{harvtxt|Banach|1932}}, spaces of [[continuous function]]s, of [[differentiable function]]s or of [[integrable function]]s, and all the Banach spaces used in functional analysis for the next forty years, contain some ℓ<sup>''p''</sup> or ''c''<sub>0</sub>. Also, new attempts in the early '70s<ref>see {{harvtxt|Lindenstrauss|1970}}, {{harvtxt|Milman|1970}}.</ref> to promote a geometric theory of Banach spaces led to ask <ref>The question is formulated explicitly in {{harvtxt|Lindenstrauss|1970}}, {{harvtxt|Milman|1970}}, {{harvtxt|Lindenstrauss|1971}} on last page. {{harvtxt|Lindenstrauss|Tzafriri|1977}}, p. 95, say that this question was "''a long standing open problem going back to Banach's book''" ({{harvtxt|Banach|1932}}), but the question does not appear in Banach's book. However, Banach compares the ''linear dimension'' of ℓ<sup>''p''</sup> to that of other classical spaces, a somewhat similar question.</ref> whether or not ''every'' infinite dimensional Banach space has a subspace isomorphic to some ℓ<sup>''p''</sup> or to ''c''<sub>0</sub>.
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| The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the [[Distortion problem|arbitrarily distortable]] space of Schlumprecht ({{harvtxt|Schlumprecht|1991}}), on which depend [[Timothy Gowers|Gowers']] solution to Banach's hyperplane problem<ref>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.</ref> and the Odell–Schlumprecht solution to the [[distortion problem]]. Also, several results of Argyros et al.<ref>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.</ref> are based on [[Ordinal number|ordinal]] refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.<ref>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<sub>∞</sub>-space that solves the scalar-plus-compact problem''", Acta Mathematica (2011) 206: 1-54.</ref> | |
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| == Tsirelson's construction ==
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| On the vector space ℓ<sup>∞</sup> of bounded scalar sequences {{nowrap| ''x'' {{=}}}} {{nowrap|{''x''<sub>''j'' </sub>} <sub>''j''∈'''N'''</sub>}}, let ''P''<sub>''n''</sub> denote the [[linear operator]] which sets to zero all coordinates ''x''<sub>''j''</sub> of ''x'' for which ''j'' ≤ ''n''.
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| A finite sequence <math>\{x_n\}_{n=1}^N</math> of vectors in ℓ<sup>∞</sup> is called ''block-disjoint'' if there are natural numbers <math>\textstyle \{a_n, b_n\}_{n=1}^N</math> so that <math>a_1 \leq b_1 < a_2 \leq b_2 < \ldots \leq b_N</math>, and so that <math>(x_n)_i=0</math> when <math>i<a_n</math> or <math>i>b_n</math>, for each ''n'' from 1 to ''N''.
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| The [[unit ball]]  ''B''<sub>∞</sub>  of ℓ<sup>∞</sup> is [[Compact space|compact]] and [[Metrization theorem|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''<sub>∞</sub>  satisfying the following two properties:<ref name="Conditions">conditions '''b''', '''c''', '''d''' here are conditions (3), (2) and (4) respectively in {{harvtxt|Tsirel'son|1974}}, and '''a''' is a modified form of condition (1) from the same article.</ref>
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| :'''a.''' For every integer  ''j''  in '''N''', the [[unit vector]] ''e''<sub>''j''</sub> and all multiples <math>\lambda e_j</math>, for |λ| ≤ 1, belong to ''K''.
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| :'''b.''' For any integer ''N'' ≥ 1, if <math>\textstyle (x_1,\dots,x_N)</math> is a block-disjoint sequence in ''K'', then <math>\textstyle{{1\over2}P_N(x_1 + \cdots + x_N)}</math> belongs to ''K''.
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| This set ''K'' satisfies the following stability property:
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| :'''c.''' Together with every element ''x'' of ''K'', the set ''K'' contains all vectors ''y'' in ℓ<sup>∞</sup> such that |''y''| ≤ |''x''| (for the pointwise comparison).
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| It is then shown that ''K'' is actually a subset of ''c''<sub>0</sub>, the Banach subspace of ℓ<sup>∞</sup> consisting of scalar sequences tending to zero at infinity. This is done by proving that
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| :'''d:''' for every element ''x'' in ''K'', there exists an integer ''n'' such that 2 ''P''<sub>''n''</sub>(''x'') belongs to ''K'', | |
| and iterating this fact. Since ''K'' is pointwise compact and contained in ''c''<sub>0</sub>, it is [[weak topology|weakly compact]] in ''c''<sub>0</sub>. Let ''V'' be the closed [[convex hull]] of ''K'' in ''c''<sub>0</sub>. It is also a weakly compact set in ''c''<sub>0</sub>. It is shown that ''V'' satisfies '''b''', '''c''' and '''d'''. | |
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| The Tsirelson space ''T''* is the Banach space whose [[unit ball]] is ''V''. The unit vector basis is an [[Schauder basis|unconditional basis]] for ''T''* and ''T''* is reflexive. Therefore, ''T''* does not contain an isomorphic copy of ''c''<sub>0</sub>. The other ℓ<sup>p</sup> spaces, 1 ≤ ''p'' < ∞, are ruled out by condition '''b'''.
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| == Properties ==
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| The Tsirelson space {{mvar|T*}} is [[reflexive space|reflexive]] ({{harvtxt|Tsirel'son|1974}}) and [[finitely universal space|finitely universal]], which means that for some constant {{nowrap|{{mvar|C}} ≥ 1}}, the space {{mvar|T*}} contains {{mvar|C}}-isomorphic copies of every finite dimensional normed space, namely, for every finite dimensional normed space {{mvar|X}}, there exists a subspace {{mvar|Y}} of the Tsirelson space with [[Banach-Mazur compactum|multiplicative Banach–Mazur distance]] to {{mvar|X}} less than {{mvar|C}}. Actually, every finitely universal Banach space contains ''almost-isometric'' copies of every finite dimensional normed space,<ref>this is because for every {{mvar|n}}, {{mvar|C}} and ε, there exists {{mvar|N}} such that every {{mvar|C}}-isomorph of ℓ<sup>∞</sup><sub>{{mvar|N}}</sub> contains a {{nowrap|(1 + ε)}}-isomorph of ℓ<sup>∞</sup><sub>''n''</sub>, 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 {{nowrap|(1 + ε)}}-embeds in ℓ<sup>∞</sup><sub>{{mvar|n}}</sub> when {{mvar|n}} is large enough.</ref> meaning that {{mvar|C}} can be replaced by {{nowrap|1 + ε}} for every {{nowrap|ε > 0}}. Also, every infinite-dimensional subspace of {{mvar|T*}} is finitely universal. On the other hand, every infinite-dimensional subspace in the dual {{mvar|T}} of {{mvar|T*}} contains almost isometric copies of <math>\scriptstyle{\ell^1_n}</math>, the {{mvar|n}}-dimensional ℓ<sup>1</sup>-space, for all {{mvar|n}}.
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| The Tsirelson space {{mvar|T}} is [[distortion problem|distortable]], but it is not known whether it is [[distortion problem|arbitrarily distortable]].
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| The space {{mvar|T*}} is a ''minimal'' Banach space.<ref>see {{harvtxt|Casazza|Shura|1989}}, p. 54.</ref> This means that every infinite dimensional Banach subspace of {{mvar|T*}} contains a further subspace isomorphic to {{mvar|T*}}. Prior to the construction of {{mvar|T*}}, the only known examples of minimal spaces were ℓ<sup>''p''</sup> and {{mvar|c}}<sub>0</sub>. The dual space {{mvar|T}} is not minimal.<ref>see {{harvtxt|Casazza|Shura|1989}}, p. 56.</ref>
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| The space {{mvar|T*}} is [[Polynomially reflexive space|polynomially reflexive]].
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| == Derived spaces ==
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| The '''symmetric Tsirelson space''' ''S''(''T'') is polynomially reflexive and it has the [[approximation property]]. As with ''T'', it is reflexive and no ℓ<sup>''p''</sup> space can be embedded into it.
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| Since it is symmetric, it can be defined even on an [[uncountable]] supporting set, giving an example of non-[[separable space|separable]] polynomially reflexive [[Banach space]].
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| ==See also==
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| * [[Distortion problem]]
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| * [[Sequence space]], [[Schauder basis]]
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| {{more footnotes|date=February 2012}}
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| ==Notes==
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| {{reflist|colwidth=auto}}
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| == References ==
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| *{{citation
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| | last = Tsirel'son | first = B. S. | author-link = Boris Tsirelson
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| | doi = 10.1007/BF01078599
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| | journal = Functional Analysis and Its Applications
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| | mr = 0350378
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| | pages = 138–141
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| | title = 'Not every Banach space contains an imbedding of ℓ<sup>''p''</sup> or ''c''<sub>0</sub>
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| | volume = 8
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| | year = 1974}}.
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| *{{citation
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| | last1 = Figiel | first1 = T.
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| | last2 = Johnson | first2 = W. B. | author2-link = William B. Johnson (mathematician)
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| | journal = Compositio Mathematica
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| | mr = 0355537
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| | pages = 179–190
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| | title = A uniformly convex Banach space which contains no ℓ<sup>''p''</sup>
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| | url = http://www.numdam.org/item?id=CM_1974__29_2_179_0
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| | volume = 29
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| | year = 1974}}.
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| *{{citation
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| | first=Stefan |last=Banach |authorlink=Stefan Banach
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| | url=http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10
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| | title=Théorie des opérations linéaires
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| | publication-place=Warszawa
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| | publisher=Subwencji Funduszu Kultury Narodowej
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| | year=1932
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| | series=Monografie Matematyczne
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| | volume=1
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| | zbl=0005.20901}}.
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| *{{citation
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| | last1 = Casazza | first1 = Peter G.
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| | last2 = Shura | first2 = Thaddeus J.
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| | isbn = 3-540-50678-0
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| | location = Berlin
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| | mr = 981801
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Mathematics
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| | title = Tsirelson's Space
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| | volume = 1363
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| | year = 1989}}.
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| *{{citation
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| | title = Handbook of the Geometry of Banach Spaces
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| | editor1-last = Johnson | editor1-first = William B.
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| | editor2-last = J. Lindenstrauss | editor2-first = Joram
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| | publisher = Elsevier
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| | volume = 1, 2
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| | publication-date = 2001, 2003}}.
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| *{{Citation
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| | last=Lindenstrauss | first=Joram |author-link = Joram Lindenstrauss
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| | title=Some aspects of the theory of Banach spaces
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| | journal = Advances in Math
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| | volume = 5
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| | pages = 159–180
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| | year = 1970}}.
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| *{{Citation
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| | last=Lindenstrauss | first=Joram |author-link = Joram Lindenstrauss
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| | title=The geometric theory of the classical Banach spaces
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| | journal = Actes du Congrès Intern. Math., Nice 1970
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| | pages = 365–372
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| | year = 1971}}.
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| *{{citation
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| | last1=Lindenstrauss | first1=Joram |author1-link = Joram Lindenstrauss
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| | last2=Tzafriri | first2=Lior
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| | isbn = 3-540-08072-4
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| | location = Berlin
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| | publisher = Springer-Verlag
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| | series = Ergebnisse der Mathematik und ihrer Grenzgebiete
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| | title=Classical Banach Spaces I, Sequence Spaces
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| | volume = 92
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| | year=1977}}.
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| *{{Citation
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| | last=Milman | first = V. D. | author-link = Vitali Milman
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| | title=Geometric theory of Banach spaces. I. Theory of basic and minimal systems
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| | language = Russian
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| | journal= Uspehi Mat. Nauk
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| | volume = 25 no. 3
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| | year = 1970
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| | pages=113–174
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| }}. English translation in Russian Math. Surveys 25 (1970), 111-170.
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| *{{Citation
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| | last= Schlumprecht | first= Th.
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| | title= An arbitrary distortable Banach space
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| | mr=1177333
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| | year=1991
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| | journal= Israel Journal of Mathematics
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| | issn=0021-2172
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| | volume=76
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| | pages=81–95}}.
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| ==External links==
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| * [http://www.tau.ac.il/~tsirel/Research/myspace/remins.html Boris Tsirelson's reminiscences on his web page]
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| [[Category:Banach spaces]]
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