Weak Convergence

May 2010
8
0
Hi

The question is:
Find an example of \(\displaystyle 1\leq p,q < \infty\) and a sequence \(\displaystyle (x_n)\) which converges strongly to 0 in \(\displaystyle l^p\), and weakly but not strongly in \(\displaystyle l^q\).

Any help would be greatly appreciated.
 

Opalg

MHF Hall of Honor
Aug 2007
4,039
2,789
Leeds, UK
Hi

The question is:
Find an example of \(\displaystyle 1\leq p,q < \infty\) and a sequence \(\displaystyle (x_n)\) which converges strongly to 0 in \(\displaystyle l^p\), and weakly but not strongly in \(\displaystyle l^q\).
If p < q and \(\displaystyle x_n\to0\) in \(\displaystyle l^p\) then \(\displaystyle x_n\to0\) in \(\displaystyle l^q\). So for your example you need to take q < p.

Suppose for example that q = 2 and p = 4. Define the k'th coordinate of \(\displaystyle x_n\) by \(\displaystyle x_n(k) = \begin{cases}n^{-1/2}&\text{if $n+1\leqslant k\leqslant 2n$,}\\ 0&\text{otherwise.}\end{cases}\)

Then \(\displaystyle \|x_n\|_4 = n^{-1/4}\), so \(\displaystyle x_n\to0\) strongly in \(\displaystyle l^4\). But \(\displaystyle \|x_n\|_2 = 1\), so \(\displaystyle x_n\not\to0\) strongly in \(\displaystyle l^2\). I'll leave it to you to figure out why \(\displaystyle x_n\to0\) weakly in \(\displaystyle l^2\).
 
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