What does $\psi \in \mathscr{L}^2(\mathbb{R}^2)^7$ mean?

Question

I'm aware that $\mathscr{L}^2(S)$ usually means the $L^p$-norm over some set in the context of metric spaces. So I guess that the first part "$\mathscr{L}^2(\mathbb{R}^2)$" means a set of $L^2$-norms over the $\mathbb{R}^2$. But when it comes to the $(\mathbb{R}^2)^7$ I could'nt figure it out what the seven in the exponent means. I found it in this paper in section "3.2.2. Adaptive Filtering" in the first paragraph.

Answer 1

That's a footnote/endnote. If you scroll down to the end of the paper, you'll see that endnote 7 says "$\mathscr{L}^2$ is the square integrable space". It's just confusing formatting.

Answer 2

For all purposes, $(\Bbb{R}^2)^7=\Bbb{R}^{14}$ (they are isomorphic).

The reason for writing it this way is probably physical. For example if you had $7$ particles in a plane. Each particle is in $\Bbb{R}^2$, and there are $7$ of them.

In practice, this is just $\mathscr{L}^2(\Bbb{R}^{14})$.