(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In [tex]l_1[/tex] (each element is a sequence of numbers such that the series converges absolutely) show that no pair of the norms [tex]||\cdot||_1, \, ||\cdot||_2, \, ||\cdot||_{\infty} [/tex] are equivalent norms.

2. Relevant equations

[tex]||x||_1=\sum_{i=1}^{\infty}|x_i|[/tex]

[tex]||x||_2=\left(\sum_{i=1}^{\infty}|x_i|^2\right)^{\frac{1}{2}}[/tex]

[tex]||x||_{\infty}=\mbox{ max}\{|x_i|:i=1,2,\ldots\}[/tex]

3. The attempt at a solution

I realize I need to show:

[tex]\dfrac{||x||_1}{||x||_2}=\infty \text{ or } 0\quad \dfrac{||x||_1}{||x||_{\infty}}=\infty \text{ or } 0\quad \dfrac{||x||_2}{||x||_{\infty}}=\infty \text{ or } 0[/tex]

but I am having problems finding such a sequence and showing this is true.

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# Homework Help: Equivalent Norms

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