In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the Tsirelson space is the first example of a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
in which neither an
ℓ ''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 notation
[see 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 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 them. The word is ...
, 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 continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s, of
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
s or of
integrable function
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s, 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
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 pre ...
which sets to zero all coordinates ''x''
''j'' of ''x'' for which ''j'' ≤ ''n''.
A finite sequence
of vectors in ℓ
∞ is called ''block-disjoint'' if there are natural numbers
so that
, and so that
when