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 be a normed vector space and let be a sequence in that converges weakly to some in :
That is, for every continuous linear functional the continuous dual space of
Then there exists a function and a sequence of sets of real numbers
such that and
such that the sequence defined by the convex combination
converges strongly in to ; that is
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 mathematics
- 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.
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