1. ## Norms in R^m

Define $||x||=\sqrt{(x1)^2 +...+(xm)^2}$ in R^m, and $||x||_{p} = \sqrt[p]{|x1|^{p} +...+|x|^{p}}$. I need to show that for every m and p there are scalars c and C such that, for every x in R^m, $c||x|| \leq ||x||_{p} \leq C||x||$.

My only thought is to try to find the limit of $C||x||-||x||_{p}$ as each coordinate goes to infinity. If it converges the I think I'm done, no? But I don't see how to do it.

2. However, being in the first part of an analysis course, the notion of a limit is not yet defined so this is probably not allowed.

3. Let me write this a little better (now that I can do it on my computer rather than an iPhone).

Define $||\text{\bf x}|| = \sqrt{(x_{1})^{2} + ... + (x_{m})^{2}}$ for $\text{\bf x} \in \mathbb{R}^{m}$, and $||\text{\bf x}||_{p} = \sqrt[p]{|x_{1}|^{p} + ... + |x_{m}|^{p}}$. Then prove what was said above: for any $m, p \geq 1$, there are scalars c, C > 0 such that for any $\text{\bf x} \in \mathbb{R}^{m}$ we have $c||\text{\bf x}|| \leq ||\text{\bf x}||_{p} \leq C||\text{\bf x}||$.

4. Originally Posted by ragnar
Define $||\text{\bf x}|| = \sqrt{(x_{1})^{2} + ... + (x_{m})^{2}}$ for $\text{\bf x} \in \mathbb{R}^{m}$, and $||\text{\bf x}||_{p} = \sqrt[p]{|x_{1}|^{p} + ... + |x_{m}|^{p}}$. Then prove what was said above: for any $m, p \geq 1$, there are scalars c, C > 0 such that for any $\text{\bf x} \in \mathbb{R}^{m}$ we have $c||\text{\bf x}|| \leq ||\text{\bf x}||_{p} \leq C||\text{\bf x}||$.
Hint: Let $\|{\bf x}\|_\infty = \max\{|x_1|,|x_2|,\ldots,|x_m|\}$. Show that $\|{\bf x}\|_\infty\leqslant\|{\bf x}\|_p\leqslant\|{\bf x}\|_1\leqslant m\|{\bf x}\|_\infty.$ Notice also that $\|{\bf x}\|=\|{\bf x}\|_2$.

5. Okay, I see how to prove the inequality of your hint. What I don't see is how this contributes to the assignment. I also see that $||\text{\bf x}||_{\infty} \leq ||\text{\bf x}||_{2} \leq ||\text{\bf x}||_{1}$, but I'm still stuck.

6. I think I may have part of it. If I multiply $||\text{\bf x}||$ by $\sqrt{m}$ then for whichever term is the max, I'll have $m$ many of it, thus meaning that the stuff under the radical will be closer to being demonstrably larger than $||\text{\bf x}||_{p}$.

7. Originally Posted by ragnar
Okay, I see how to prove the inequality of your hint. What I don't see is how this contributes to the assignment. I also see that $||\text{\bf x}||_{\infty} \leq ||\text{\bf x}||_{2} \leq ||\text{\bf x}||_{1}$, but I'm still stuck.
Once you have proved those inequalities, you're practically there:

$\|{\bf x}\|_p \leqslant m\|{\bf x}\|_\infty \leqslant m\|{\bf x}\|_2,$

$\|{\bf x}\|_2\leqslant m\|{\bf x}\|_\infty \leqslant m\|{\bf x}\|_p$ and hence $m^{-1}\|{\bf x}\|_2\leqslant \|{\bf x}\|_p.$

Thus $m^{-1}\|{\bf x}\| \leqslant \|{\bf x}\|_p \leqslant m\|{\bf x}\|.$

8. Ah, got it... But WOW I'm amazed I was expected to have figured this all out without even the hint you gave.

Thank you very much for your help! That was very good.