Locally convex lattice

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A real topological vector space $E$ that is simultaneously a vector lattice and whose topology is a locally convex topology, while the mappings of $E \times E$ into $E$ defined by $$ (x,y) \mapsto \sup\{x,y\} \,,\ \ \ (x,y) \mapsto \inf\{x,y\} \ \ \text{for}\ x,y \in E \,, $$ are continuous. General questions in the theory of locally convex lattices are the following: The study of the connections between topological properties and order properties; in particular, the topological properties of bands and positive cones in a locally convex lattice and connections between lattice properties and topological properties of completeness in a locally convex lattice. The study of properties of the strong dual of a locally convex lattice and properties of the imbedding of a locally convex lattice $E$ into its second dual. The construction of a theory of extension of positive functionals and linear mappings between locally convex lattices.

The most important example of a locally convex lattice is a Banach lattice.


[1] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)
[2] M.M. Day, "Normed linear spaces" , Springer (1958)
[3] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)



[a1] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
[a2] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
[a3] H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
How to Cite This Entry:
Locally convex lattice. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article