While trying to understand a proof of equivalence of norms for $H^k(\mathbb{R}^n)$ (Fourier Transforms) I came across a possible inconsistency in the multi-index notation. Can somebody please clarify it for me? It is surprisingly hard to find it explained clearly anywhere.
Suppose $x \in \mathbb{R}^n$ and that $\alpha = (\alpha_1,...,\alpha_n)$ is a multi-index. In lines with the multi-index notation we have $x^{\alpha} = x_{1}^{\alpha_1}...x_{n}^{\alpha_n}$. But then what is $|x^{\alpha}|^2$? If we were to be consistent then $|x^{\alpha}|^2 = |x_{1}^{\alpha_1}...x_{n}^{\alpha_n}|^2$, but that's not what is meant in this scenario, right? Here $|x^{\alpha}|^2 = \sum_{i=1}^n (x_i^{\alpha_i})^2$ or something like this, as with $|x|^2 = \sum_{i=1}^n x_i^2 $
See this question for some reference: Sobolev spaces fourier norm equivalence or Evans book on PDEs, Theorem 8, page 297, which is exactly about proving this result.
I am perhaps completely in the dark about the problem and simply there is here a mixture of a notational confusion plus me not understanding the proof. I would have commented on the referenced question to ask for some insight, but being a completely new user I am not allowed to. I would really appreciate any feedback (also about asking questions here as it is my first attempt). Thank you!