Approximation property
Encyclopedia
In mathematics
, 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.
Every Hilbert space
has this property. There are, however, Banach space
s which do not; Enflo published the first counterexample in an 1973 article. However, a lot of work in this area was done by Grothendieck (1955).
Later many other counterexamples were found. The space of bounded operator
s on
does not have the approximation property (Szankowski). The spaces 
for 
and 
(see Sequence space
) have closed subspaces that do not have the approximation property.
is said to have the approximation property, if, for every compact set 
and every 
, there is an operator 
of finite rank so that 
, for every 
.
Some other flavours of the AP are studied:
Let
be a Banach space and let 
. We say that 
has the 
-approximation property (
-AP), if, for every compact set 
and every 
, there is an operator 
of finite rank so that 
, for every 
, and 
.
A Banach space is said to have bounded approximation property (BAP), if it has the
-AP for some 
.
A Banach space is said to have metric approximation property (MAP), if it is 1-AP.
A Banach space is said to have compact approximation property (CAP), if in the
definition of AP an operator of finite rank is replaced with a compact operator.
has the AP (we can use the projections associated to the base as the
's in the definition), thus a lot of spaces with the AP can be found. For example, the 
spaces
, or the symmetric Tsirelson 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...
, a 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...
is said to have the approximation property (AP), if every compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
is a limit of finite-rank operators. The converse is always true.
Every Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
has this property. There are, however, 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...
s which do not; Enflo published the first counterexample in an 1973 article. However, a lot of work in this area was done by Grothendieck (1955).
Later many other counterexamples were found. The space of bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on
does not have the approximation property (Szankowski). The spaces for and (see Sequence spaceSequence 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 numbers to the field K of real or complex numbers...
) have closed subspaces that do not have the approximation property.
Definition
A Banach spaceBanach 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...
is said to have the approximation property, if, for every compact set and every , there is an operator of finite rank so that , for every .Some other flavours of the AP are studied:
Let
be a Banach space and let . We say that has the -approximation property (-AP), if, for every compact set and every , there is an operator of finite rank so that , for every , and .A Banach space is said to have bounded approximation property (BAP), if it has the
-AP for some .A Banach space is said to have metric approximation property (MAP), if it is 1-AP.
A Banach space is said to have compact approximation property (CAP), if in the
definition of AP an operator of finite rank is replaced with a compact operator.
Examples
Every space with a Schauder basisSchauder basis
In mathematics, a Schauder basis or countable basis is similar to the usual 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...
has the AP (we can use the projections associated to the base as the
's in the definition), thus a lot of spaces with the AP can be found. For example, the spacesLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
, or the symmetric Tsirelson space
Tsirelson space
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...
.