I am reading the book "Classical Fourier Analysis by Grafakos" and did encounter some questions about the Fourier Transformation.
Disclaimer: I am not that far in Analysis to be able to claim to understand Analysis in $\mathbb{R}^n$ very good. For that reason I am trying to first understand Fourier Analysis in $\mathbb{R}$ before $\mathbb{R}^n$. But since the book is writen with $\mathbb{R}^n$ in mind I am trying to rewrite the definitions for $\mathbb{R}$.
$\frac{\partial f}{\partial x_i}=:\partial_i f$
Let $\alpha=(\alpha_1,...,\alpha_n) \in \mathbb{N}^n$ and let $\partial1^{\alpha}f:=\partial_1^{\alpha_1}...\partial_n^{\alpha_n}f$
The in the chapter about Fourier transform the book starts with defining the class of Schwartz functions on $\mathbb{R}^n$ by
Definition: A $C^{\infty}$ complex valued function $f$ on $\mathbb{R}^n$ is called a Schwartz function if for every pair of multi indices $\alpha$ and $\beta$ there exists a positive constant $C_{\alpha,\beta}$ such that: $\rho_{\alpha,\beta}(f)=sup_{x \in \mathbb{R}^n}|x^{\alpha}\partial^{\beta}f(x)|=C_{\alpha,\beta}< \infty$.
I think that for $n=1$ the definiton is: ...if for every pair of Indices $\alpha, \beta \in \mathbb{N_0}$ there exists a posiitive constant $C_{\alpha,\beta}$ such that: $\rho_{\alpha,\beta}(f)=sup_{x \in \mathbb{R}}|x^{\alpha}\frac{d }{d x}f(x)|=C_{\alpha,\beta}< \infty$
Is that correct?
Edit 1: My first version was a little bit wrong, as I saw in the comments, so here the corrected version:
$\rho_{\alpha,\beta}(f)=sup_{x \in \mathbb{R}}|x^{\alpha}\frac{d^{\beta} }{d^{\beta} x}f(x)|=C_{\alpha,\beta}< \infty$