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 ${\displaystyle (X,\|\,\cdot \,\|)}$ be a normed vector space and let ${\displaystyle \left(u_{n}\right)_{n\in \mathbb {N} }}$ be a sequence in ${\displaystyle X}$ that converges weakly to some ${\displaystyle u_{0}}$ in ${\displaystyle X}$:

$${\displaystyle u_{n}\rightharpoonup u_{0}{\mbox{ as }}n\to \infty .}$$

That is, for every continuous linear functional ${\displaystyle f\in X^{\prime },}$ the continuous dual space of ${\displaystyle X,}$

$${\displaystyle f\left(u_{n}\right)\to f\left(u_{0}\right){\mbox{ as }}n\to \infty .}$$

Then there exists a function ${\displaystyle N:\mathbb {N} \to \mathbb {N} }$ and a sequence of sets of real numbers

$${\displaystyle \left\{\alpha (n)_{k}:k=n,\dots ,N(n)\right\}}$$

such that ${\displaystyle \alpha (n)_{k}\geq 0}$ and

$${\displaystyle \sum _{k=n}^{N(n)}\alpha (n)_{k}=1}$$

such that the sequence ${\displaystyle \left(v_{n}\right)_{n\in \mathbb {N} }}$defined by the convex combination

$${\displaystyle v_{n}=\sum _{k=n}^{N(n)}\alpha (n)_{k}u_{k}}$$

converges strongly in ${\displaystyle X}$ to ${\displaystyle u_{0}}$; that is

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

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 – 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.