Mazur's lemma
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Wikipedia : http://en.wikipedia.org/wiki/Mazur's_lemmaIn 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 u_{0} 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 (v_{n})_{n∈N} defined by the convex combination
converges strongly in X to u_{0}, 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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