I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less than $F$ is exact at the point $u\in V$ if $l(u)=F(u)$, where $F$ is a mapping of normed space $V$ into $\mathbb R$. Then, necessarily:
$l$ will have the form $$ l(v)=\left<v-u,u^*\right>+F(u); $$
$l$ is maximal: its constant term is the greatest possible, when ce: $$ F(u)-\left<u,u^*\right>=-F^*(u^*), $$ where $u^*\in V^*$ is the dual of $V$ and $F^*$ is the conjugate (dual) function of $F$.
I can understand 1. and 2. is true, but I can't see why it is "necessarily" true. Any help is really welcome!