Mazur's lemma

The set of Mazur (after Stanisław Mazur ) is a set of functional analysis that indicates a relationship between the weak and the strong convergence. From the definitions it follows immediately that every strongly convergent sequence converges weakly, while the weak convergence is not a sufficient criterion for the strong convergence. The set of Mazur notes now that one can construct from convex combinations of terms of a weakly convergent sequence of a strongly convergent sequence.

Wording of the sentence

Is a normed vector space and against weak convergent sequence. Then there exists a sequence of convex combinations of (ie ), so that strong ( ie with respect to the norm of ) to converge.

Sketch of proof

You need two results from functional analysis: (1) locally convex topological vector spaces are complete and closed convex sets weak. (2 ) In addition to the standard degree of convex sets is convex again.

Every normed vector space is a locally convex topological vector space.

Consider the set of all convex combinations so the (so-called convex hull ). Their standard degree is again convex ( 1 ), so is the closed convex hull of the weakly closed (2). Now, as a weak limit of elements from the closed convex hull of an element of this. This limit of a sequence of convex combinations must be the.

Source

• Dirk Werner: Functional Analysis. 6, corrected edition, Springer -Verlag, Berlin 2007, ISBN 978-3-540-72533-6, page 108

• Functional Analysis
• Set ( mathematics)