In mathematics, Mazur's lemma is a result in the theory of normed vector spaces. It shows that any weakly convergent sequence in a normed space has a sequence of convex combinations of its members that converges strongly to the same limit, and is used in the proof of Tonelli's theorem.
Mazur's theorem — Let be a normed vector space and let be a sequence converges weakly to some .
Then there exists a sequence made up of finite convex combination of the 's of the form