# Weak Convergence

#### markwolfson16900

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
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 $$\displaystyle l^2$$.

• markwolfson16900