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Thread: Weak Convergence

  1. #1
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    Weak Convergence

    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.
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  2. #2
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    Quote Originally Posted by markwolfson16900 View Post
    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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