Ordered vector space

Summary

An ordered vector space is a real vector space with a compatible partial order. The order is preserved by adding a vector and by multiplying by a nonnegative scalar. Such an order is the same as a positive cone, the set of vectors that are at least $0$. Spaces of real functions, of sequences and of self-adjoint operators carry natural orders of this kind. Ordered vector spaces are the setting of positive linear maps, and Riesz spaces are the case where the order is a lattice.