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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 $$M$$.

##### MSC:
 46A40 Ordered topological linear spaces, vector lattices 46E05 Lattices of continuous, differentiable or analytic functions 06D10 Complete distributivity
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