One can show that a given vector field  can be decomposed as
Different than the usual Helmholtz decomposition, the
Helmholtz–Leray decomposition of  is unique (up to an
additive constant for  ). Then we can define  as
The Leray projector is defined similarly on function spaces other than the Schwartz space, and on different domains with different boundary conditions. The four properties listed below will continue to hold in those cases.