Linear systems with locally integrable trajectories

The locally integrable function space L-1(loc) is made a module over the ring of proper rational functions. Using this module structure, the notion of relative dimension is defined naturally for every linear subspace in (L-1(loc))(q). It is shown that the property of having finite relative dimension together with the property of having sufficiently many smooth trajectories and the evident property of differentiation-invariance characterize the weak solutions sets of linear constant coefficient differential systems among all linear subspaces. (C) 2008 Elsevier Inc. All rights reserved.