Let $u \in C_0^\infty (\mathbb R)$, $v(x) := u(x) e^{-x^2 /2} $. And define the norm as $ \| u \|_1^2 = \int_{\mathbb R} | u' (x) |^2 e^{-x^2} dx + \int_{\mathbb R} | u(x) |^2 e^{-x^2} dx $ Then I want to prove that $ \| u \|_1^2 = \int_{\mathbb R} ( | v' (x) |^2 + x^2 | v(x)|^2 ) dx $
I think this is not trivial by just using the definition of the norm above. $C_0^\infty $ means that $C^\infty$ functions with a compact support. And I have one more question.
If the condition $u \in C_0^\infty ( \mathbb R) $ changes to "$\| u \|_1^2 < \infty$", then does this still hold?