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 .

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.