CHARACTERIZATIONS OF BOUNDED APPROXIMATION PROPERTIES

New necessary and sufficient conditions for Banach spaces to have bounded approximation properties are established, which are easier to check than known ones. Also using these it is shown that for a Banach space $X$, the dual $X^{*}$ has the bounded approximation property if and only if $X$ has the bounded approximation property approximated by the weak adjoint operator topology, and if $X^{*}$ has the bounded weak approximation property, then $X$ has the bounded weak approximation property approximated by the weak adjoint operator topology.