what is $|x|$ for $x \in \mathbb{R}^{n}$?

Question

What is |x| given by in dimension n? What is |x| when x lies in the set of real numbers to the power n. This is needed in order to find the laplacian of u when u is equal to an equation involving |x|. What is $|x|$ when $x \in \mathbb{R}^{n}$

Answer 1

What is typically meant is the frobenius norm $|x|=\sqrt{\sum_{i=1}^n x_i^2}$

Answer 2

if $x=(x_1,x_2,...x_n)$ is an element of $R^n$ then you can take the Euclidian norm given by $|x|=\sqrt{x_1^2+x_2^2+...x_n^2}$.

Answer 3

What $|x|$ means in a general normed space is the norm of $x$, where a norm is a function from $X\times X\to \Bbb R$. The most typical norm in $\Bbb R^n$ is the euclidean norm where $$|x|=\sqrt{\sum_{i=1}^n x_i^2}$$ This is however not all the norms possible to concieve, however many norms are equivalent to it.

Answer 4

The three are equivalent and you can use any of them:

$|x|= \sqrt{ x_1^2 + \cdots x_n^2}$,

$|x|= \max\{|x_1|, \cdots, |x_n|\}$,

$ |x|= |x_1|+ \cdots+ |x_n|$.