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

AbstractIf 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, and2.H ordered by K: (i) is a vector lattice; (ii) ||x||=|||x|||,∀x∈H, and (iii) x∈K,y−x∈K imply ||x||⩽||y||