Friday, 2 May 2014

real analysis - Showing that a sequence converges in norm

I am trying to prove the following:



Let X be a normed linear space satisfying the property: {xn},{yn}X, we have
xn=yn=1,xn+yn2xnyn0.



If {zn}X converges to zX weakly (meaning limnf(zn)=z for all fX) and znz, then znz0.



Here is what I am trying to do:




I can consider {z} as a sequence in X. I want to show that zn+z2. Well, since znz=1, then since zn+zzn+z, then limnzn+z2z=2.



I can't figure out how to possibly show that limnzn+z2. How would I even incorporate the weak convergence assumption? Any help would be greatly appreciated! Thank you.

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