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Lattices of Lipschitz functions. (English) Zbl 0797.46007
Let $M$ be a metric space. We observe that $\text{Lip}(M)$ has a striking lattice structure: its closed unit ball is lattice-complete and completely distributive. This motivates further study into the lattice structure of $\text{Lip}(M)$ and its relation to $M$. We find that there is a nice duality between $M$ and $\text{Lip}(M)$ (as a lattice). We also give an abstract classification of all normed vector lattices which are isomorphic to $\text{Lip}(M)$ for some $M$.
Reviewer: N.Weaver (Berkeley)

46A40Ordered topological linear spaces, vector lattices
46E05Lattices of continuous, differentiable or analytic functions
06D10Complete distributivity of lattices
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