Mazur's lemma

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

Mazur's theorem — Let $(X,\lVert \cdot \rVert )$ be a normed vector space and let $\left\{x_{j}\right\}_{j\in \mathbb {N} }\subset X$ be a sequence converges weakly to some $x\in X$.

Then there exists a sequence $\left\{y_{k}\right\}_{k\in \mathbb {N} }\subset X$ made up of finite convex combination of the $x_{j}$'s of the form

$$y_{k}=\sum _{j\geq k}\lambda _{j}^{(k)}x_{j}$$

such that $y_{k}\to x$ strongly that is $\lVert y_{k}-x\rVert \to 0$.

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 – theorem in mathematicsPages displaying wikidata descriptions as a fallback
• 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. p. 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. p. 6.