Mazur's lemma

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 ${\displaystyle (X,\|\,\cdot \,\|)}$ be a Banach 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 ${\displaystyle X}$:

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

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

such that ${\displaystyle \alpha (n)_{k}\geq 0}$ and such that the sequence ${\displaystyle \left(v_{n}\right)_{n\in \mathbb {N} }}$defined by the convex combination converges strongly in ${\displaystyle X}$ to ${\displaystyle u_{0}}$; that is

See also[]

• Banach–Alaoglu theorem – The closed unit ball in the dual of a normed vector space is compact in the weak* topology
• 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. p. 350. ISBN 0-387-00444-0.