$\mathbb{K}=\mathbb{R}$. Let $X$ be a reflexive Banach space, $M\subset X$ nonempty, closed and convex. Let $J\in X'$and $B:X\times X \rightarrow \mathbb{R}$ be symmetric nonegative bilinear form. For each $u\in X$ assume $B(u,.)$ is continuous w.r.t weak convergence ($B(u,v_{n})\rightarrow B(u,v), v_{n}\rightharpoonup v$).
$B(u,u)\ge c||u||^2$ for all $u\in M$. Define $E: X \rightarrow \mathbb{R}$
by $E(u)=B(u,u)-J(u)$
let $(u_{k})_{k\in \mathbb{N}}$ be a sequence in $M$ such that $E(u_{k})\rightarrow inf_{u\in M}E(u)$ as $k \rightarrow \infty$. Prove that $u_{k}$ possesses a subsequence which converges weakly to some element $u_{0}\in M$ and show $E(u_{0})= infE(u)$
Now assume that $M$ is closed linear subspace of $X$. Let $u_{0}\in M$ be as in (1.). Prove that $ 2B(u_{0},h)=J(h)$ for all $ h \in M$
Can someone help me with these questions i don't know how to do it .