Given , I have to prove that , where denotes the p-norm, with . Since I've already proved the original Minkowski inequality, a simple induction estabilishes the finite case, i.e., , .
So, what I've thought is, since is a Banach space, to suppose that , which would give us that converges in , by completeness. Hence, the desired inequality follow by making in , but I'm not completely sure about this.
Any help will be appreciated.