i have this space $H=\lbrace u\in AC([0,+\infty), u(0)=u(+\infty)=0, \sqrt{p}u'\in L^2((0,+\infty))\rbrace$ where $p>0$ and $\displaystyle\frac1p\in L^1$
how to see that the quantity: $\displaystyle||u||^2=\displaystyle\int_0^{+\infty} p(t)u'^2(t) dt$ is a norm ?
i have $||u||=0$ impliese that $u'(t)=0$ why $u=0$ ?
Thank you