Ratio of $\|\cdot\|$ and $\|\cdot\|_{\infty}$ on $\mathbb{R}^2$

Question

I have the following question from an old examination paper in Real Analysis:

On $\mathbb{R}^2$, $\|\cdot\|_{\infty}$ is defined by $\left\|\left(x,y\right)\right\|_{\infty}:=\max\left\{\left|x\right|,\left|y\right|\right\}$. Let $\|\cdot\|$ denote another norm on $\mathbb{R}^2$. Show that $$ \inf\left\{\frac{\left\|\left(x,y\right)\right\|}{\left\|\left(x,y\right)\right\|_{\infty}}:\left(x,y\right)\in\mathbb{R}^2\setminus\left\{\left(0,0\right)\right\}\right\}>0 $$

My initial thoughts were that this is a very simple question and that as $\|\cdot\|$ and $\|\cdot\|_{\infty}$ are only $0$ with $\left(x,y\right)=\left(0,0\right)$, and are always non negative then this identity was obvious. Sadly the answer provided is a lot longer and states that the identity is the same as $$ \inf\left\{\left\|\left(x,y\right)\right\|:\left(x,y\right)\in\mathbb{R}^2,\left\|\left(x,y\right)\right\|_{\infty}=1\right\}=J\geq0 $$ then using this to prove that $J\not=0$ and therefore $J>0$.

Could somebody please explain

• Why my intuition is incorrect

• How the two inequalities are the same

Answer 1

Consider $\large\inf\{\frac{\|(x,y)\|}{\|(x,y)\|_\infty}\}$ there is an associated point $(x^*,y^*) \in \mathbb{R}^2$ such that $\|(x^*,y^*)\|_\infty=1$

and $(x,y)=\|(x,y)\|_\infty(x^*,y^*)$, i.e. $\large(x^*,y^*)=\frac{(x,y)}{\|(x,y)\|_\infty}$

Thus $\large\inf\{\frac{\|(x,y)\|}{\|(x,y)\|_\infty}|(x,y)\in\mathbb{R}^2\setminus (0,0)\}$

$=\large\inf\{\frac{\|\|(x,y)\|_\infty(x^*,y^*)\|}{\|\|(x,y)\|_\infty(x^*,y^*)\|_\infty}|(x^*,y^*)\in\mathbb{R}^2\setminus (0,0)\}$

$=\large\inf\{\frac{\|(x^*,y^*)\|}{\|(x^*,y^*)\|_\infty}|(x,y)\in\mathbb{R}^2\setminus (0,0)\}$

$=\large\inf\{\|(x^*,y^*)\||(x^*,y^*)\in\mathbb{R}^2|\|(x^*,y^*)\|_\infty=1\}=J\ge 0$

We know that $J\ge 0$ since $\|\cdot\|$ is a well defined norm.

And we know that $J\ne 0$, since $\|(x^*,y^*)\|_\infty=1\Rightarrow (x^*,y^*)\ne (0,0)$

So again since $\|\cdot\|$ is a well defined norm, $J\ne 0$.

Answer 2

Firstly, there is the possibility that you have a sequence of points $(x_n,y_n)$ with $||(x_n,y_n)||=n$, but $||(x_n,y_n)||_{\infty}=n^2$, so the ratio is $1/n$, and the infimum of $\{1/n\}$ is zero.

Secondly, for any vector $(x,y)$ in the first inequality, scale it down by $\sqrt{||(x,y)||_{\infty}}$. Both the norms scale down by $||(x,y)||_{\infty}$ and you get the second inequality.