A bounded projection on a Banach space 
is a bounded operator 
satisfying 
. Its range and kernel give
the direct decomposition 
for every 
. Both subspaces are closed:
the range is the kernel of the continuous
map 
.
A bounded projection onto a subspace 
acts as the identity on 
.
The norm-one case is used in the definition of an injective von Neumann algebra (Pisier 2020).
