The Lipschitz-free space of a pointed metric space 
is the closed vector
space span of the evaluation functionals 
in the continuous
dual of the real 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 can be approximated uniformly on every compact set by finite-rank linear operators. The bounded approximation property additionally requires one finite upper bound on their operator norms.
