Mazur's lemma

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 which converges weakly to some

Then there exists a sequence

made up of finite convex combination of the

's of the form

λ

strongly that is