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

Mazur's theorem  Let be a normed vector space and let be a sequence converges weakly to some .

Then there exists a sequence made up of finite convex combination of the 's of the form

such that strongly 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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