Approximation property

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.

Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955).

Later many other counterexamples were found. The space ${\displaystyle {\mathcal {L}}(H)}$ of bounded operators on an infinite-dimensional Hilbert space ${\displaystyle H}$ does not have the approximation property.^[2] The spaces ${\displaystyle \ell ^{p}}$ for ${\displaystyle p\neq 2}$ and ${\displaystyle c_{0}}$ (see Sequence space) have closed subspaces that do not have the approximation property.

Definition

A locally convex topological vector space X is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank.^[3]

For a locally convex space X, the following are equivalent:^[3]

1. X has the approximation property;
2. the closure of ${\displaystyle X^{\prime }\otimes X}$ in ${\displaystyle \operatorname {L} _{p}(X,X)}$ contains the identity map ${\displaystyle \operatorname {Id} :X\to X}$;
3. ${\displaystyle X^{\prime }\otimes X}$ is dense in ${\displaystyle \operatorname {L} _{p}(X,X)}$;
4. for every locally convex space Y, ${\displaystyle X^{\prime }\otimes Y}$ is dense in ${\displaystyle \operatorname {L} _{p}(X,Y)}$;
5. for every locally convex space Y, ${\displaystyle Y^{\prime }\otimes X}$ is dense in ${\displaystyle \operatorname {L} _{p}(Y,X)}$;

where ${\displaystyle \operatorname {L} _{p}(X,Y)}$ denotes the space of continuous linear operators from X to Y endowed with the topology of uniform convergence on pre-compact subsets of X.

If X is a Banach space this requirement becomes that for every compact set ${\displaystyle K\subset X}$ and every ${\displaystyle \varepsilon >0}$, there is an operator ${\displaystyle T\colon X\to X}$ of finite rank so that ${\displaystyle \|Tx-x\|\leq \varepsilon }$, for every ${\displaystyle x\in K}$.

Related definitions

Some other flavours of the AP are studied:

Let ${\displaystyle X}$ be a Banach space and let ${\displaystyle 1\leq \lambda <\infty }$. We say that X has the ${\displaystyle \lambda }$-approximation property (${\displaystyle \lambda }$-AP), if, for every compact set ${\displaystyle K\subset X}$ and every ${\displaystyle \varepsilon >0}$, there is an operator ${\displaystyle T\colon X\to X}$ of finite rank so that ${\displaystyle \|Tx-x\|\leq \varepsilon }$, for every ${\displaystyle x\in K}$, and ${\displaystyle \|T\|\leq \lambda }$.

A Banach space is said to have bounded approximation property (BAP), if it has the ${\displaystyle \lambda }$-AP for some ${\displaystyle \lambda }$.

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 subspace of an arbitrary product of Hilbert spaces possesses the approximation property.^[3] In particular,
  • every Hilbert space has the approximation property.
  • every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.^[3]
  • every nuclear space possesses the approximation property.
• Every separable Frechet space that contains a Schauder basis possesses the approximation property.^[3]
• Every space with a Schauder basis has the AP (we can use the projections associated to the base as the ${\displaystyle T}$'s in the definition), thus many spaces with the AP can be found. For example, the ${\displaystyle \ell ^{p}}$ spaces, or the symmetric Tsirelson space.

References

Bibliography

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