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

Let $(X, \, \,\cdot\,\, )$ be a normed vector space and let $\left(u_n\right)_$ be a sequence in $X$ that converges weakly to some $u_0$ in $X$:
$u_n \rightharpoonup u_0 \mbox n \to \infty.$

That is, for every continuous linear functional $f \in X^,$ the continuous dual space of $X,$
$f\left(u_n\right) \to f\left(u_0\right) \mbox n \to \infty.$

Then there exists a function $N : \N \to \N$ and a sequence of sets of real numbers
$\left\$
such that $\alpha(n)_k \geq 0$ and
$\sum_^ \alpha(n)_ = 1$
such that the sequence $\left(v_n\right)_$defined by the convex combination
$v_n = \sum_^ \alpha(n)_ u_$
converges strongly in $X$ to $u_0$; that is
$\left\, v_n - u_0\right\, \to 0 \mbox n \to \infty.$

See also

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References

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{{Convex analysis and variational analysis

Banach spaces
Theorems involving convexity
Theorems in functional analysis
Lemmas in analysis
Compactness theorems