  Let G be an amenable group and V be a finite dimensional vector space. Gromov
pointed out that the von Neumann dimension of linear subspaces of l^2(G;V)
(with respect to G) can be obtained by looking at a growth factor for a
dynamical (pseudo-)distance. This dynamical point of view (reminiscent of
metric entropy) does not requires a Hilbertian structure. It is used in this
article to associate to a $\Gamma$-invariant linear subspaces Y of l^p(G;V) a
real positive number dim_{l^p} Y (which is the von Neumann dimension when p=2).
By analogy with von Neumann dimension, the properties of this quantity are
explored to conclude that there can be no injective G-equivariant linear map of
finite-type from l^p(G;V) -> l^p(G; V') if dim V > dim V'. A generalization of
the Ornstein-Weiss lemma is developed along the way.
