Does it make sense notation to say $L^{2}$ norm on $\mathbb{R}^n$?

Question

I was reading something and it says let $\| x\|_2$ be the $L^2$ norm on $\mathbb{R}^n$. This is a minor point but does it actually make sense to call it $L^2$ norm here? I thought maybe $\ell^2$ norm maybe more appropriate? (maybe it does not matter at all?) And this would be the same for $\ell^{\infty}$ and $L^{\infty}$ as well perhaps?

Answer 1

Yes, $\mathbb{R^n}$ with the $p$-norm is a special $L^p$ space with the counting measure on $\{1,...,n\}$: Let $x \in \mathbb{R}^n$. Then we have that $$\int |x|^p \mathrm{d}\mu=\sum_{i=1}^n |x(i)|^p$$ Because we can identify every element of $\mathbb{R}^n$ as a function from $\{1,...,n\}$ to $\mathbb{R}$.