Definition
A Banach space is said to have the approximation property, if, for every compact set and every, there is an operator of finite rank so that, for every .
Some other flavours of the AP are studied:
Let be a Banach space and let . We say that has the -approximation property (-AP), if, for every compact set and every, there is an operator of finite rank so that, for every, and .
A Banach space is said to have bounded approximation property (BAP), if it has the -AP for some .
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.
Read more about this topic: Approximation Property
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