What does the symbol $\|\ \|^2_2$ mean?

Question

If the distance is given by: $$\|X_m-X_n\|^2_2.$$

Could anyone precisely explain the meaning of the symbol $\|\ \|^2_2$ ?
Thanks, in advance.

Answer 1

$\|x\|_2$ means the $2$-norm, which is $$\sqrt{\sum_{i=1}^n x_i^2}$$

while the $2$ on top, is its square, so both result in: $$\|x\|^2_2=\sum_{i=1}^n x_i^2.$$

Overall, it means the sum of square of the components.

Answer 2

In general, $|| \cdot ||_2$ is called the $L^2$-norm, which is a functional from the function space $L^2(\Omega,{\cal A}, \mu)$ to $\Bbb R$ defined by $f \mapsto (\int |f|^2 \,{\rm d}\mu)^{1/2}$, provided that the integral exists. This is probably the case when you see $$||X_m - X_n||_2$$ since random variables are usually denoted with capital letters in probability-theory.

In particular, when $\Omega = \Bbb{N}$, $\mu$ is the counting measure, we have $||f||_2 = (\int |f|^2 \,{\rm d}\mu)^{1/2} = (\sum_n |f_n|^2)^{1/2}$. This is the case if you actually mean $$||x_m-x_n||_2$$ a norm of vector.

Remark: It's hard to distinguish $X$ from $x$ in your drawing. That's the reason why I promote the use of $\rm \LaTeX$ in all levels of math writing.