Mazur's lemma

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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)nN be a sequence in X that converges weakly to some u0 in X:

That is, for every continuous linear functional f in X, the continuous dual space of 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)nN defined by the convex combination

converges strongly in X to u0, i.e.

References

  • 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. ISBN 0-387-00444-0. 
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