# Approximation property

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 ${\mathcal {L}}(H)$ of bounded operators on an infinite-dimensional Hilbert space $H$ does not have the approximation property. 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.
For a locally convex space X, the following are equivalent:
* X has the approximation property;
* the closure of $X^{\prime }\otimes X$ in $\operatorname {L} _{p}(X,X)$ contains the identity map $\operatorname {Id} :X\to X$;
* $X^{\prime }\otimes X$ is dense in $\operatorname {L} _{p}(X,X)$;
* for every locally convex space Y, $X^{\prime }\otimes Y$ is dense in $\operatorname {L} _{p}(X,Y)$;
* for every locally convex space Y, $Y^{\prime }\otimes X$ is dense in $\operatorname {L} _{p}(Y,X)$;
where $\operatorname {L} _{p}(X,Y)$ 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 $K\subset X$ and every $\varepsilon >0$, there is an operator $T\colon X\to X$ of finite rank so that $\|Tx-x\|\leq \varepsilon$, for every $x\in K$.

## Related definitions

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 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.