# Proof of additivity of the positive linear functional $\phi^+$ on a vector/Banach lattice that will be $\phi\vee0$.

For context, this is used in defining $$\phi\vee0$$ in a proof that the dual of a vector lattice is a vector lattice.

Given a linear functional $$\phi$$ on a vector lattice $$V$$, define $$\phi^+$$ on $$V^+$$ as $$\phi^+: v \mapsto \sup\{\phi(u) : 0.

How can I show $$\phi^+(v+w) = \phi^+(v) + \phi^+(w)\ \forall v,w \in V^+$$ ?

Thanks!