I want to show that the Schwartz functions are dense in
$$\left\{f \in L^2; \int |x|^2 \left|f(x)\right|^2 dx + \int |\xi|^2 \left|\hat{f}(\xi)\right|^2 d \xi < \infty\right\}$$
where the norm is given by $$\left\lVert f\right\rVert_{L^2}^2 = \int (1+|x|^2) |f(x)|^2 dx + \int \left(1+|\xi|^2\right) \left|\hat{f}(\xi)\right|^2 d \xi.$$
If we would only have one summand, then this would be a $H^1-$ Sobolev summand norm and for this space I know that this is true, but how can I show this for this more general norm?