Definition of energy space

Question

In the literature and also in this question the energy space is defined as $$E:=\{f\in S'(\mathbb R^d)\ |\ \|\nabla f\|_{L^2}+\|xf\|_{L^2}<\infty\}.$$

I understand every part of this definition except: Where does $x$ come from?

Answer 1

The formulation is not quite exact. This is what it means: $$\|xf\|_{L^2}^2=\int_{\mathbb R^d}(x\cdot f(x),x\cdot f(x))_{\mathbb C^d}dx=\int_{\mathbb R^d}\|x\|_{\mathbb R^d}^2|f(x)|^2dx,$$ where $(\cdot,\cdot)_{\mathbb C^d}$ is the Euclidean scalar product in $\mathbb C^d$ and $\|\cdot\|_{\mathbb R^d}$ is the Euclidean norm in $\mathbb R^d$.