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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 \(M\).

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