# 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 $\displaystyle{ (X, \|\,\cdot\,\|) }$ be a normed vector space and let $\displaystyle{ \left(u_n\right)_{n \in \N} }$ 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 $\displaystyle{ N : \N \to \N }$ 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 $\displaystyle{ \left(v_n\right)_{n \in \N} }$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 $\displaystyle{ \left\|v_n - u_0\right\| \to 0 \mbox{ as } n \to \infty. }$