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

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

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

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

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.