Tsirelson space
Encyclopedia
In mathematics
, Tsirelson space T is an example of a reflexive
Banach space
in which neither an l p space
nor a c0 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
for the dual of the Tsirelson space.
denote the linear operator which sets all coordinates 
, 
to zero.
We call a sequence
block-disjoint, if for each 
there are natural numbers 
and 
, so that 
when 
or 
. Also, 
.
Define these four properties for a set
:
We define
as the space with unit ball 
, where 
is an absolutely convex
weakly
compact set, for which (1)-(4) hold true. It may be noted that a set with the given properties exists, but is not unique.
and finitely universal. Also, every infinite-dimensional subspace is finitely universal.
is polynomially reflexive
and it has the approximation property
. As with
, it is reflexive and no 
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Tsirelson space T is an example of a reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
in which neither an l p space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
nor a c0 space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
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
for the dual of the Tsirelson space.Construction
Let denote the linear operator which sets all coordinates , to zero.We call a sequence
block-disjoint, if for each there are natural numbers and , so that when or . Also, .Define these four properties for a set
:
-
is contained in the unit ball of 
. Every unit vector 
is in 
. -
(pointwise comparison) - For any
, let 
be a block-disjoint sequence in 
, then 
. -
.
We define
as the space with unit ball , where is an absolutely convexConvex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
weakly
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
compact 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 reflexiveReflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
and finitely universal. Also, every infinite-dimensional subspace is finitely universal.
Derived spaces
The symmetric Tsirelson space is polynomially reflexivePolynomially reflexive space
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....
and it has the approximation property
Approximation property
In mathematics, a Banach space is said to have the approximation property , if every compact operator is a limit of finite-rank operators. The converse is always true....
. As with
, it is reflexive and no 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
Polynomially reflexive space
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....
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
.