Let H be a Hopf algebra and M a representation or a corepresentation o
f H. In this paper we study semiinvariants of M. This notion generaliz
es the known concept of weight spaces in the context of representation
s of Lie algebras. Our best results are attained for pointed Hopf alge
bras, and semiinvariants which are related to the coradical filtration
of H.