I have the following boundary value problem, where $k, q \in \mathbb{R}$ are given.
$$-u'' + ku' +qu = f,\qquad u(0)=u(1)= 0$$
What I want to prove is that the bilinear form, associated with this BVP, is continuous and also coercive.
By definition we have that, a bilinear forma $(\cdot,\cdot)$ on a normed linear space H is said to continuous if $\exists C< \infty$ such that $$|a(v, w)|\le C‖v‖_H‖w‖_H, \qquad\forall v, w\in H$$ So this is what I should prove in order to show it is continuous.
I tried to write the variational formulation for this is as: $$V=H^1(0,1)$$ $$a(u, v)=\int_0^1(u'v'+ku'v+quv)\ dx$$ $$F(v)=(f, v)$$
So I have that, $$|a(u,v)|\le\Bigg|\int_0^1ku'v\ dx\Bigg|+\Bigg|\int_0^1u'v'+quv\ dx\Bigg|$$ $$\le|k|||u'||_{L^2}||v||_{L^2}+\Bigg|\int_0^1u'v'+quv\ dx\Bigg|$$
Not sure how to proceed from here, can anyone help? Also if I made any mistakes, please tell me.