Let B be a reflexive Banach space, X a locally convex space and T : B → X (not necessarily bounded) linear transformation. A necessary and sufficient condition is obtained so that for a given v ∈ X there is a solution for the equation T u = v . This result is used to discuss the existence of an L p -weak solution of D u = v where D is a differential operator with smooth coefficients and v ∈ L p .