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.

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