x 8 X = L2(J,V) , dx/dt 6 X* = L2(J,V*), t 6 J = [0,T], f 6 L2(J,V*), has at l e a s t one weak s o l u t i o n x 6 X, Ix[ ! R = al l f l x . . Thereby, we have obtained the well known existence of at least one solution z 6 X to the Cauchy problem with initial data x(0) = 0 for the linear evolution equation, by sole topological considerations. Natural extensions make this existence statement essentially equivalent to others proved independently (see[5], and the end of no. 5 for some of the extensions mentioned in a more particular situation). An analytical existence proof is given, for instance, in Lions [18], where also uniqueness is proved. As the reader may have noticed, the llnearity of A plays here no role. We sha~l consider a quasilinear problem in no. 6.