I am learning about functional analysis at the moment and I have difficulties grasping the connection to integral equations or differential equations. For simplicity, let us consider the following problem
Let $U\subset \mathbb R^n$ be open and bounded, $f\in L^2(U)$ and $b\in L^\infty(U, \mathbb R^n)$. For $u,w\in H^1_0(U)$ define $B(u,w):=\int_U (\nabla u\cdot\nabla w+b\cdot \nabla uw)dx$. Suppose that for $v\in H^1_0(U)$ with $B(v,w)=0$ for all $w\in H^1_0(U)$ it follows that $v=0$.
Show that there exists a unique $u\in H^1_0(U)$ such that $B(u,w)=\int_U fwdx$ for all $w\in H^1_0(U)$.
How do I approach such a Problem with Fredholm Alternative? How do I locate the compact operator? Are the subtleties I need to watch out for?