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: 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.
 * 1) X has the approximation property;
 * 2) the closure of $$X^{\prime} \otimes X$$ in $$\operatorname{L}_p(X, X)$$ contains the identity map $$\operatorname{Id} : X \to X$$;
 * 3) $$X^{\prime} \otimes X$$ is dense in $$\operatorname{L}_p(X, X)$$;
 * 4) for every locally convex space Y, $$X^{\prime} \otimes Y$$ is dense in $$\operatorname{L}_p(X, Y)$$;
 * 5) for every locally convex space Y, $$Y^{\prime} \otimes X$$ is dense in $$\operatorname{L}_p(Y, X)$$;

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.