On strongly convergent combinations of a weakly convergent sequence in a Banach space
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.
Statement of the lemma
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
such that strongly that is .
Renardy, Michael & Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 350. ISBN0-387-00444-0.
Ekeland, Ivar & Temam, Roger (1976). Convex analysis and variational problems. Studies in Mathematics and its Applications, Vol. 1 (Second ed.). New York: North-Holland Publishing Co., Amsterdam-Oxford, American. p. 6.