Characterization of a Hilbert vector lattice by the metric projection onto its positive cone

If H is a real Hilbert space, K is a closed, generating cone therein and PK is the metric projection onto K, then the following two conditions 1 and 2 are equivalent: 1. (i) PK is isotone: y−x∈K⇒PK(y)−PK(x)∈K and (ii) PK is subadditive: PK(x)+PK(y)−PK(x+y)∈K, ∀x, y∈H, and 2. H ordered by K: (i) is a vector lattice; (ii) ||x||=|| |x| ||, ∀x∈H, and (iii) x∈K, y−x∈K imply ||x||⩽||y||.