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



If \left\{z_n\right\} \subseteq X converges to z\in X weakly (meaning \displaystyle \lim_{n\rightarrow \infty} f(z_n)=z for all f\in X^*) and \|z_n\| \rightarrow \|z\|, then \|z_n-z\|\rightarrow 0.



Here is what I am trying to do:




I can consider \left\{z \right\} as a sequence in X. I want to show that \|z_n+z\|\rightarrow 2. Well, since \|z_n\| \rightarrow \|z\|=1, then since \|z_n+z\|\leq\|z_n\|+\|z\|, then \displaystyle \lim_{n\rightarrow \infty} \|z_n+z\| \leq 2\|z\|=2.



I can't figure out how to possibly show that \displaystyle \lim_{n\rightarrow \infty} \|z_n+z\| \geq 2. How would I even incorporate the weak convergence assumption? Any help would be greatly appreciated! Thank you.

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