# Integral Equation and Fredholm Alternative

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?

Sincerely, slin0