Prove l1 and l-infinity norm equality.

Asked Nov 13 '22 at 13:35

Active Nov 13 '22 at 14:09

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I struggle with a simple problem which is to show that L1 norm is equivalent to L-infinity norm ($\ \mathbb{R}^n$). I know that this holds $$\ ||x||_1 \ge ||x||_\infty $$ so the C2 will be equal to 1.

I can't find the C1 so that $$\ ||x||_\infty \ge C1||x||_1$$

edited Nov 13 '22 at 14:09

asked Nov 13 '22 at 13:35

PiotrWj

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On which space? $\Bbb R^n$? if so, $|x|1\le n|x|\infty.$ – Anne Bauval Nov 13 '22 at 13:53
@AnneBauval Yes $\ R^n$. How would the proof of that look like? – PiotrWj Nov 13 '22 at 14:02
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1/2: then edit your question to specify it. 2/2: follow the link to the duplicate or simply: sum up the $n$ inequalities $|x_k|\le|x|_\infty.$ – Anne Bauval Nov 13 '22 at 14:03
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No, the duplicate is simply the post I linked to (and its accepted answer). But the hint in my previous comment is sufficient. – Anne Bauval Nov 13 '22 at 14:16

Prove l1 and l-infinity norm equality.

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