Approximation property

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 spaces which do not; Per Enflo published the first counterexample in a 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 operators on $\ell^2$ does not have the approximation property (Szankowski). The spaces $\ell^p$ for $p\neq 2$ and $c_0$ (see Sequence space) have closed subspaces that do not have the approximation property.

Definition

A Banach space $X$ is said to have the approximation property, if, for every compact set $K\subset X$ and every $\varepsilon>0$ , there is an operator $T\colon X\to X$ of finite rank so that $\|Tx-x\|\leq\varepsilon$ , for every $x\in K$ .

Some other flavours of the AP are studied:

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

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 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 a lot of spaces with the AP can be found. For example, the $\ell^p$ spaces, or the symmetric Tsirelson space.

References

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