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Lattices of Lipschitz functions. (English) Zbl 0797.46007
Let M be a metric space. We observe that 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 Lip(M) and its relation to M. We find that there is a nice duality between M and Lip(M) (as a lattice). We also give an abstract classification of all normed vector lattices which are isomorphic to Lip(M) for some M.

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