Mazur's lemma - Alchetron, The Free Social Encyclopedia

In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach 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 (X, || ||) be a Banach space and let (u_n)_n∈N be a sequence in X that converges weakly to some u₀ in X:

u n ⇀ u 0 as n → ∞ .

That is, for every continuous linear functional f in X^∗, the continuous dual space of X,

f ( u n ) → f ( u 0 ) as n → ∞ .

Then there exists a function N : N → N and a sequence of sets of real numbers

{ α ( n ) k | k = n , … , N ( n ) }

such that α(n)_k ≥ 0 and

∑ k = n N ( n ) α ( n ) k = 1

such that the sequence (v_n)_n∈N defined by the convex combination

v n = ∑ k = n N ( n ) α ( n ) k u k

converges strongly in X to u₀, i.e.

∥ v n − u 0 ∥ → 0 as n → ∞ .

References

Mazur's lemma Wikipedia

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