# 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.[1]

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 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.[2] If X is a Banach space the 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.

## Examples

• Every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.[3]
• Hence every nuclear space possesses the approximation property.
• Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property.[4]
• Every separable Frechet space that contains a Schauder basis possesses the approximation property.[5]
• 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

1. ^ Megginson, Robert E. An Introduction to Banach Space Theory p. 336
2. ^ Schaefer p. 108
3. ^ Schaefer p. 110
4. ^ Schaefer p. 109
5. ^ Schaefer p. 115