Let us now consider the following abstract minimization problem (M): Find $u \in V$ $$F(u)=min_{v \in V} F(v)$$ where $$F(v)=\frac{1}{2} a(v,v)- L(v),$$ and consider also the following abstract variatonal problem (V): Find $u \in V$ $$a(u,v)=L(v) \forall v \in V.$$ Theorem 2.1 The problems (M) and (V) arc equivalent, ie, $u\in V$ satisfies (M) if and only if u satisfies (V).
Can u help me with the proof of this?(with the reverse implicity,(V)-->(M))
Use $$ F(u+h)=\frac12 a(u,u)+a(u,h)+\frac12a(h,h)-L(u)-L(h)=\frac12a(h,h)+F(u) $$ $F(u)$ is a constant and $a(h,h)$ is non-negative...