Set $$W=\Big\{ u:[0,+\infty )\to \mathbb{R} \; \text{is absolutely continuous and}\; u'\in L^2 [0,+\infty)\Big\},$$ where $q$ is a non-zero constant. Assume that $$X=\Big\{ u\in W:\int_0^{+\infty}(|u'(t)|^2+q^2 |u(t)|^2)dt<+\infty\Big\}$$ with the inner product $$\langle u,v\rangle:=\int_0^{+\infty}(u'(t)v'(t)+q^2u(t)v(t))dt$$ which induces the norm $$\|u\|:=\Big( \int_0^{+\infty}(|u'(t)|^2+q^2 |u(t)|^2)dt\Big)^{\frac{1}{2}}$$.
How can we prove that $X$ is a reflexive Banach space? This space is called the Sobolev space in the literature. Is there any proof for that? Thanks in advance.