  We introduce the concept of a $\lambda$-Hopf algebra as a Hopf algebra
obtained as the partial smash product algebra of a Hopf algebra and its base
field, and show that every Hopf algebra is a $\lambda$-Hopf algebra.
  Moreover, a method to compute partial actions of a given Hopf algebra on its
base field is developed and, as an application, we exhibit all partial actions
of such type for some families of Hopf algebras.
