How can I show the space with norm $\|f\| = \left(\int_{\mathbb R^n} (1+|x|^2)^m |f(x)|^2 dx\right)^{1/2}$ is complete?

Question

Let $m\in \mathbb R$. How can I show that $X = \{f:\mathbb R^n \to \mathbb C| \|f\|<\infty\}$ with the norm $$ \|f\| = \left(\int_{\mathbb R^n} (1+|x|^2)^m |f(x)|^2 dx\right)^{1/2}, $$ is complete?

Answer 1

Hint:

Suppose $(f_n)$ is a Cauchy sequence in $X$.

Find an equivalent Cauchy sequence $(g_n)$ in $L^2$.

Use a well-known result to find $g\in L^2$, such that $g_n\to g$ in $L^2$.

Find $f$ in $X$, related to $g$, such that $f_n\to f$ in $X$.