# Locally convex lattice

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.

#### References

[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) |

#### Comments

#### References

[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: http://encyclopediaofmath.org/index.php?title=Locally_convex_lattice&oldid=39415