Approximation property

File:MazurGes.jpg

The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanisław Mazur (left) in 1936.

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 ${\ell }^p$ for $p\ne 2$ 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:

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

where ${\mathrm{L}}_p(X,Y)$ 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 $\epsilon >0$, there is an operator $T:X\to X$ of finite rank so that $\Vert Tx-x\Vert \le \epsilon$, for every $x\in K$.

Related definitions

Some other flavours of the AP are studied:

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

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

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. 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 ${\ell }^p$ spaces, or the symmetric Tsirelson space.

Bibliography

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