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Let $V$ be a normed space with two norms $f$ and $g$. Prove that these statements are equivalent:

$\{z_n\}$ tends to 0 as $n → ∞$ in $f\;\; $ $\forall \{z_n\}\in V$

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  • $\begingroup$ write explicitly what means "$\|.\|_1,\|.\|_2$ are equivalent" $\endgroup$
    – reuns
    Nov 16, 2016 at 14:32
  • $\begingroup$ This is the same as saying the open-ball $\|x\|_1 < 1$ is contained in some open ball $\|x\|_2 < c$, and conversely. Now what happens if $\|x_n\|_1 \to 0$ iff $\|x_n\|_2 \to 0$ ? $\endgroup$
    – reuns
    Nov 16, 2016 at 15:12
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    $\begingroup$ Don't vandalize your questions, $\endgroup$ Nov 16, 2016 at 20:38

1 Answer 1

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Assume there is no $\alpha$ such that $\Vert x\Vert_1\le \alpha\Vert x\Vert_2$ for all $x$. So for every $n\in\Bbb N$ there is some $x_n$ such that $\Vert x_n\Vert_1>n\Vert x_n\Vert_2$. Let

$$z_n=\frac{x_n}{\sqrt n \Vert x_n\Vert_2}$$ so $\Vert z_n\Vert_2=\frac1{\sqrt n}\xrightarrow{n\to\infty}0$ wheras $\Vert z_n\Vert_1\ge \sqrt n\xrightarrow{n\to\infty}+\infty$.

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