Let X be a (real or complex) normed space, T the topology on X induced by the norm, D a nonempty subset of X which is not dense in X, and M:= fT 0 j T 0 is a locally convex topology on X; T 0 µ T ; D T 0 = Xg; where D T 0 stands for the closure of D in the topological space (X; T 0). We first give a method for obtaining elements T 0 in M in the case D is a linear subset of X and the domain of a linear operator A: D ! X. Afterwards we present an application to m-accretive operators.