How do we show that $||x||_r \le ||x||_s$?

Asked Feb 20 '23 at 01:40

Active Feb 20 '23 at 01:40

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Consider the normed spaced $\mathbb{R}^n$. Then how do I show that $||x||_r \le ||x||_s\ $ if $\ 1\le s \le r \le \infty$?

So far I was trying to use Holder's inequality but made no progress at all. So I have nothing to show here. Any help would be appreciated. Thanks.

asked Feb 20 '23 at 01:40

Itachi

Pehaps this may help: https://math.stackexchange.com/questions/4094/how-do-you-show-monotonicity-of-the-ellp-norms – JayP Feb 20 '23 at 01:51
You can get proper double norm bars by using \| instead of ||. – joriki Feb 20 '23 at 03:44
1

This is very elementary. No need for Holder. Assume that RHS $\leq 1$ and show that LHS $\leq 1$. Then finish the proof. – Kavi Rama Murthy Feb 20 '23 at 05:03

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