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 in contains the identity map ;
- is dense in ;
- for every locally convex space Y, is dense in ;
- for every locally convex space Y, is dense in ;
where 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 and every , there is an operator of finite rank so that , for every .
- 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 's in the definition), thus many spaces with the AP can be found. For example, the spaces, or the symmetric Tsirelson space.
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