Estimate with L^2 Norm of gradient

Question

For given $u,v\in H_{0}^{1}$,$b_{i}\in L^{\infty}$ i have to show that $$\sum \int |b_{i}uv_{x_{i}}|\leq \|b\|_{L^{\infty}}\|u\|_{L^{2}}\ \|\nabla v\|_{L^{2}}$$ holds. I think it is obvious that I have to use the hölder inequality but the L2 Norm of the gradient does not appear that way.