The Lipschitz-free space of a pointed metric space 
is the closed vector
space span of the evaluation functionals 
in the dual
normed space of the real-valued Lipschitz
functions that vanish at 
, equipped with their Lipschitz
constant as norm. It is a real Banach
space, denoted 
. Its canonical evaluation
vectors satisfy 
and 
.
The approximation property means that the identity operator can be approximated uniformly on every compact set by finite-rank bounded operators. The bounded approximation property additionally requires one finite upper bound on their operator norms.
