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 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