New necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

Keywords:

approximation property, quasi approximation property, weak approximation property approximation property, quasi approximation property, weak approximation property

MSC Classifications:

46B28, 46B10 show english descriptions Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]
Duality and reflexivity [See also 46A25] 46B28 - Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]
46B10 - Duality and reflexivity [See also 46A25]

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