Converse nonimplication

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:

$u_n\rightharpoonup u_0\text{ as }n\to \mathrm{\infty }.$

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

$f(u_n)\to f(u_0)\text{ as }n\to \mathrm{\infty }.$

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.

$\Vert v_n-u_0\Vert \to 0\text{ as }n\to \mathrm{\infty }.$

References

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