On the transversality conditions and their genericity

In this note we review some results on the transversality conditions for a smooth Fredholm map (Formula Presented.) between two Banach spaces (Formula Presented.). These conditions are well-known in the realm of bifurcation theory and commonly accepted as “generic”. Here we show that under the transversality assumptions the sections (Formula Presented.) of the zero set of f are discrete for every (Formula Presented.) and we discuss a somehow explicit family of perturbations of f along which transversality holds up to a residual set. The application of these results to the case when f is the X-differential of a time-dependent energy functional (Formula Presented.) and (Formula Presented.) is the set of the critical points of E provides the motivation and the main example of this paper.