In Partial Differential Equations (Evans, 2nd edition) $\S$6.2.3, the author discusses about the Fredholm alternative w.r.t the second order elliptic PDE. My question may require that you are familiar with the textbook.
I want to check one assertion in the textbook: (page 323)
(22) holds if and only if $v$ is a weak solution of (12).
Here is my attempt:
Note that $(Ku, v) = (u, v) - B[L_{\gamma}^{-1} u, v]$, so $$ \begin{align*} & v - K^* v = 0 \\ \Leftrightarrow{} &(u,v) = (u, K^*v) = (Ku, v)\quad \forall u\in H_0^1(U) \\ \Leftrightarrow{} & \color{red}{B[L_{\gamma}^{-1} u, v] = 0\quad \forall u\in H_0^1(U)}. \end{align*} $$
However, by definition $v$ is a weak solution of (12) means that $B[u,v] = 0$ for all $u\in H_0^1(U)$, which is not exactly the same as the red part. What information do I miss out? How to prove the equivalence between $v - K^* v = 0$ and $B[u,v] = 0\ \forall u\in H_0^1(U)$?