Mazur's 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:

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

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

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

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

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

such that α(n)_k ≥ 0 and

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

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

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

converges strongly in X to u₀, i.e.

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

• 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.