# 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, much work in this area was done by Grothendieck (1955).

Later many other counterexamples were found. The space of bounded operators on $\displaystyle{ \ell^2 }$ does not have the approximation property. 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.

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

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 $\displaystyle{ \varepsilon\gt 0 }$, 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 $\displaystyle{ 1\leq\lambda\lt \infty }$. We say that X has the $\displaystyle{ \lambda }$-approximation property ($\displaystyle{ \lambda }$-AP), if, for every compact set $\displaystyle{ K\subset X }$ and every $\displaystyle{ \varepsilon\gt 0 }$, 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. 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 $\displaystyle{ T }$'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.