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.

[edit] 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:

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 (vn)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.

[edit] References

  • Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 350. ISBN 0-387-00444-0.