In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach 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
Let (X, || ||) be a Banach space and let (un)n∈N be a sequence in X that converges weakly to some u0 in X:
Then there exists a function N : N → N and a sequence of sets of real numbers
such that α(n)k ≥ 0 and
such that the sequence (vn)n∈N defined by the convex combination
converges strongly in X to u0, i.e.
- Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second edition ed.). New York: Springer-Verlag. p. 350. ISBN 0-387-00444-0.