# Approximation property 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, 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 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. If X is a Banach space this requirement becomes that 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.