# Math Help - Weak Convergence

1. ## Weak Convergence

Hi

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

Any help would be greatly appreciated.

2. Originally Posted by markwolfson16900
Hi

The question is:
Find an example of $1\leq p,q < \infty$ and a sequence $(x_n)$ which converges strongly to 0 in $l^p$, and weakly but not strongly in $l^q$.
If p < q and $x_n\to0$ in $l^p$ then $x_n\to0$ in $l^q$. 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 $x_n$ by $x_n(k) = \begin{cases}n^{-1/2}&\text{if n+1\leqslant k\leqslant 2n,}\\ 0&\text{otherwise.}\end{cases}$

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