We know that the following definition for Sobolev space $H^s(\mathbb{R}^n) = \{ v \in L^2(\mathbb{R}^n): (1+\xi^2)^{s/2} \hat{v}(\xi) \in L^2(\mathbb{R}^n)\}$, valid for $s\geq0$.
In these notes (https://www.math.uci.edu/~chenlong/226/Ch1Space%20(Long-Chens-MacBook-Air.local's%20conflicted%20copy%202017-10-23).pdf), at example 1.11, the author says that $\delta \in H^{-s}(\mathbb{R}^n)$ for $s>\frac{n}{2}$, and to justify this he basically put the Fourier transform of $\delta$, which is $1$, in the definition of the norm:
$$\int_{\Omega} (1+|\xi|^2)^s d\xi$$ and after integration in polars we have: $\delta \in H^{-s}(\mathbb{R}^n)$ for $s>\frac{n}{2}$. However, I don't see why this implies that $\delta$ is an element of the dual space.
What am I missing?