what is the absolute value of an absolute value of x and what is the absolute value multiplied with the same absolute value

Question

So, if I take the absolute value over the absolute value: $|x|$ does it become like this $||x||$ (I know it is something different than the Euclidian norm) or will it just remain like $|x|$

What I think:

$|x|$ $*$ $| x |$ = $x^2$ right?

Since, $|x|$= $\sqrt{x^2}$ and $\sqrt{a}$ $*$ $\sqrt{a}$ = $a$ or

Answer 1

The absolute value: $|\cdot|:\mathbb{R}\rightarrow \mathbb{R}_+$ is given by: $|x|=\begin{cases} x,\text{ if }x\geq 0\\-x\text{ if }x<0\end{cases}$. I will write: $abs(x):=|x|$. If you want to compute: $abs(abs(x))$, you know that $abs(x)\geq 0$, therefore, using the definition, $abs(abs(x))=abs(x)$. Indeed, $abs(x)^2=x^2$, and $\sqrt{x^2}=abs(x)$ (since the radical function takes nonnegative values. The euclidean norm: $\|\cdot\|:\mathbb{R}^n\rightarrow \mathbb{R}$ is defined by $\|(x_1,\dots,x_n)\|=\sqrt{x_1^2+\dots+x_n^2}$, therefore the absolute value is a particular case of the Euclidean norm, for $n=1$.

Answer 2

Yes you are right. And $||x||$ is just $|x|$ since the absolute value is always non- negative. In general you have that $|g(x)|=g(x)$ if and only if $g(x)$ is non-negative.