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 [math]\displaystyle{ (X, \|\,\cdot\,\|) }[/math] be a normed vector space and let [math]\displaystyle{ \left(u_n\right)_{n \in \N} }[/math] be a sequence in [math]\displaystyle{ X }[/math] that converges weakly to some [math]\displaystyle{ u_0 }[/math] in [math]\displaystyle{ X }[/math]: [math]\displaystyle{ u_n \rightharpoonup u_0 \mbox{ as } n \to \infty. }[/math]
That is, for every continuous linear functional [math]\displaystyle{ f \in X^{\prime}, }[/math] the continuous dual space of [math]\displaystyle{ X, }[/math] [math]\displaystyle{ f\left(u_n\right) \to f\left(u_0\right) \mbox{ as } n \to \infty. }[/math]
Then there exists a function [math]\displaystyle{ N : \N \to \N }[/math] and a sequence of sets of real numbers [math]\displaystyle{ \left\{ \alpha(n)_k : k = n, \dots, N(n) \right\} }[/math] such that [math]\displaystyle{ \alpha(n)_k \geq 0 }[/math] and [math]\displaystyle{ \sum_{k = n}^{N(n)} \alpha(n)_{k} = 1 }[/math] such that the sequence [math]\displaystyle{ \left(v_n\right)_{n \in \N} }[/math]defined by the convex combination [math]\displaystyle{ v_n = \sum_{k = n}^{N(n)} \alpha(n)_{k} u_{k} }[/math] converges strongly in [math]\displaystyle{ X }[/math] to [math]\displaystyle{ u_0 }[/math]; that is [math]\displaystyle{ \left\|v_n - u_0\right\| \to 0 \mbox{ as } n \to \infty. }[/math]
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
- 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. pp. 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. pp. 6.
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