Mazur's lemma

Short description: On strongly convergent combinations of a weakly convergent sequence in a Banach space

In mathematics, Mazur's lemma is a result in the theory of normed vector spaces. It shows that any weakly convergent sequence in a normed 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 $(X,\Vert \cdot \Vert )$ be a normed vector space and let ${\left(u_n\right)}_{n\in \mathbb{\mathbb{N}}}$ be a sequence in $X$ that converges weakly to some $u_0$ in $X$: $$u_n\rightharpoonup u_0\text{ as }n\to \mathrm{\infty }.$$

That is, for every continuous linear functional $f\in X^{\prime },$ the continuous dual space of $X,$ $$f\left(u_n\right)\to f\left(u_0\right)\text{ as }n\to \mathrm{\infty }.$$

Then there exists a function $N:\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$ and a sequence of sets of real numbers $$\{\alpha (n{)}_k:k=n,\dots ,N(n)\}$$ such that $\alpha (n{)}_k\ge 0$ and $$\sum _{k=n}^{N(n)}\alpha (n{)}_k=1$$ such that the sequence ${\left(v_n\right)}_{n\in \mathbb{\mathbb{N}}}$defined by the convex combination $$v_n=\sum _{k=n}^{N(n)}\alpha (n{)}_k{u}_k$$ converges strongly in $X$ to $u_0$; that is $$\Vert v_n-u_0\Vert \to 0\text{ as }n\to \mathrm{\infty }.$$

See also

• Banach–Alaoglu theorem – Theorem in functional analysis
• Bishop–Phelps theorem
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• James's theorem
• Goldstine theorem

References

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 350. ISBN 0-387-00444-0. 
• Ekeland, Ivar; Temam, Roger (1976). Convex analysis and variational problems. Studies in Mathematics and its Applications, Vol. 1 (Second ed.). New York: North-Holland Publishing Co., Amsterdam-Oxford, American. pp. 6.