A **cosemisimple** Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will this decomposition obey the property that the type and multiplicity of the irreducible comodules appear be the same in any decomposition.

I am sure that this should be the case but I can't see how to prove it. One thing that confuses me is the prospect of infinite multiplicity in the case of an infinite dimensional comodule.

**Note:** I previously asked this question on stackexchange, but moved it here after no response.