property of uniformly convex Banach Space

Theorem 1.

Let X be a uniformly convex Banach spaceMathworldPlanetmath. Let (xn) be a sequence in X such that limxn=x in the weak-topology (w(X,X*)) and lim supxnx.Then xn converges to x.


For x=0 the claim is obvious, so suppose that x0.The sequence (xn)n1 converges to x for w-topology xlim infxn.So let λn=max{x,xn} and we have that limλn=x. Define yn=xnλn and y=xx.Then yn converges to y in w-topology. We conclude that ylim infyn+y2. Also, y=1,yn1 so we have that limyn+y2=1. As the Banach space is uniformly convex one can easily see that limyn-y=0. Therefore xn converges to x. The proof is complete. ∎

Title property of uniformly convex Banach Space
Canonical name PropertyOfUniformlyConvexBanachSpace
Date of creation 2013-03-22 15:14:18
Last modified on 2013-03-22 15:14:18
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 20
Author georgiosl (7242)
Entry type Theorem
Classification msc 46H05