The bounded approximation property (BAP) of a Banach space 
means that there is a constant 
such that, for every compact subset 
and every 
,
there is a bounded linear
operator 
with finite-dimensional range
such that 
and 
for all 
. Omitting that uniform bound gives the approximation
property.
Every Banach space with a Schauder basis has the bounded approximation property.
