Mazur's lemma

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 (u_n)_n∈N be a sequence in X that converges weakly to some u₀ in X:

$u_{n}\rightharpoonup u_{0}{\mbox{ as }}n\to \infty .$

That is, for every continuous linear functional f in X^∗, the continuous dual space of X,

$f(u_{n})\to f(u_{0}){\mbox{ as }}n\to \infty .$

Then there exists a function N : N → N and a sequence of sets of real numbers

$\{\alpha (n)_{k}|k=n,\dots ,N(n)\}$

such that α(n)_k ≥ 0 and

$\sum _{k=n}^{N(n)}\alpha (n)_{k}=1$

such that the sequence (v_n)_n∈N defined by the convex combination

$v_{n}=\sum _{k=n}^{N(n)}\alpha (n)_{k}u_{k}$

converges strongly in X to u₀, i.e.

$\|v_{n}-u_{0}\|\to 0{\mbox{ as }}n\to \infty .$

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.