  Partial actions of Hopf algebras can be considered as a generalization of
partial actions of groups on algebras. Among important properties of partial
Hopf actions, it is possible to prove the existence of enveloping actions,
i.e., every partial Hopf action on a algebra A is induced by a Hopf action on a
algebra B that contains A as a right ideal. This globalization theorem allows
to extend several results from the theory of partial group actions to the Hopf
algebraic setting. In this article, we prove a dual version of the
globalization theorem: that every partial coaction of a Hopf algebra admits an
enveloping coaction. We also show how this works on a series of examples which
go beyond partial group actions. Finally, we explore some consequences of
globalization theorems in order to present versions of the duality theorems of
Cohen-Montgomery and Blattner-Montgomery for partial Hopf actions.
