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 $ℒ (H)$ of bounded operators on an infinite-dimensional Hilbert space $H$ does not have the approximation property. The spaces $ℓ^{p}$ for $p \neq 2$ and $c_{0}$ (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.

For a locally convex space X, the following are equivalent:

X has the approximation property;
the closure of $X^{'} \otimes X$ in $L_{p} (X, X)$ contains the identity map $Id : X \to X$ ;
$X^{'} \otimes X$ is dense in $L_{p} (X, X)$ ;
for every locally convex space Y, $X^{'} \otimes Y$ is dense in $L_{p} (X, Y)$ ;
for every locally convex space Y, $Y^{'} \otimes X$ is dense in $L_{p} (Y, X)$ ;

where $L_{p} (X, Y)$ 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 $K \subset X$ and every $ε > 0$ , there is an operator $T : X \to X$ of finite rank so that $‖ T x - x ‖ \leq ε$ , for every $x \in K$ .

Related definitions

Some other flavours of the AP are studied:

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

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

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. 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.
- every nuclear space possesses the approximation property.
Every separable Frechet space that contains a Schauder basis possesses the approximation property.
Every space with a Schauder basis has the AP (we can use the projections associated to the base as the $T$ 's in the definition), thus many spaces with the AP can be found. For example, the $ℓ^{p}$ spaces, or the symmetric Tsirelson space.

References

↑ Megginson, Robert E. An Introduction to Banach Space Theory p. 336
↑ Szankowski, Andrzej (1981). "B(H) does not have the approximation property". Acta Mathematica. 147: 89–108. doi:10.1007/BF02392870.
1 2 3 4 5 Schaefer & Wolff 1999, p. 108-115.

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[1] Megginson, Robert E. An Introduction to Banach Space Theory p. 336

[2] Szankowski, Andrzej (1981). "B(H) does not have the approximation property". Acta Mathematica. 147: 89–108. doi:10.1007/BF02392870.

[FOOTNOTESchaeferWolff1999108-115-3] 1 2 3 4 5 Schaefer & Wolff 1999, p. 108-115.