Free vector lattices over vector spaces as function lattices

We show that a free vector lattice over a real vector space $V$ can be realised canonically as a vector lattice of real-valued positively homogeneous functions on any linear subspace of its dual space that separates the points of $V$. This is used to give intuition for the known fact that the free Banach lattice over a Banach space $E$ can be realised as a Banach lattice of positively homogeneous functions on $E^\ast$. It is also applied to improve the well-known result that free vector lattices over non-empty sets can be realised as vector lattices of real-valued functions. For infinite sets, the underlying spaces for such realisations can be chosen to be smaller than the usual ones.