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$$\|X_m-X_n\|^2_2.$$ To be clear, I am not asking about the $X_m - X_n$ part. I am asking what the $\|\ \|^2_2$ thing means.

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2 Answers

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$\|x\|_2$ 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.

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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.

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