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
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)