Best approximation from the positive cone of an inner product lattice

Let E be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: i) E is a vector lattice and its norm induced by its inner product is a lattice norm. ii) The metric projection onto the positive cone E⁺ of E exists and it is both isotone and subadditive. Moreover, in this case, the best approximation to any x ∈ E from E⁺ coincides with its positive part x⁺. This result extends previous work to the non-complete setting.