In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS)
making it into vector lattice that possesses a neighborhood base at the origin consisting of solid sets.
[1] Ordered vector lattices have important applications in spectral theory.
is a vector lattice then by the vector lattice operations we mean the following maps: If
is a TVS over the reals and a vector lattice, then
is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.
is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous.
is a topological vector space (TVS) and an ordered vector space then
is called locally solid if
possesses a neighborhood base at the origin consisting of solid sets.
[1] A topological vector lattice is a Hausdorff TVS
making it into vector lattice that is locally solid.
[1] Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.
denote the set of all bounded subsets of a topological vector lattice with positive cone
Then the topological vector lattice's positive cone
[2] If a topological vector lattice
is order complete then every band is closed in
) are Banach lattices under their canonical orderings.