I'm confused about Schwartz Space. In particular I'm finding it difficult to show a function is / isn't in Schwartz space.
Definition:The Schwartz Class $\mathcal{S}(\mathbb{R}^{n})$ is the set of all functions $$f:\mathbb{R}^{n} \rightarrow \mathbb{C}$$
which satisfy $\rho_{\alpha,\beta}(f):= \sup_{x \in \mathbb{R}^{n}}|x^{\alpha}\partial^{\beta}f|< \infty$ for all multi-indices $ \alpha, \beta.$
I know that this means that the elements of the space are those continuously differentiable functions which, along with all their derivatives vanish at infinity faster than any inverse power of $x.$
Question: Why is it an inverse power of $x$ I don't see how this follows from the definition.
Question: Why is $|x|^{2} \notin \mathcal{S}(\mathbb{R}^{n})?$ I struggle to apply the definition of Schwartz space.
I know that I obviously need to show that it doesn't satisfy the definition but It's quite confusing to me.
Any help appreciated.
Thanks.