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In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order
making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.[1]
Ordered vector lattices have important applications in spectral theory.
If 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.[1]
If 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.[1]
If is a topological vector space (TVS) and an ordered vector space then
is called locally solidif
possesses a neighborhood base at the origin consisting of solid sets.[1]Atopological vector lattice is a Hausdorff TVS
that has a partial order
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.[1]
Let denote the set of all bounded subsets of a topological vector lattice with positive cone
and for any subset
, let
be the
-saturated hull of
.
Then the topological vector lattice's positive cone
is a strict
-cone,[1] where
is a strict
-cone means that
is a fundamental subfamily of
that is, every
is contained as a subset of some element of
).[2]
If a topological vector lattice isorder complete then every band is closed in
.[1]
The Lp spaces () are Banach lattices under their canonical orderings.
These spaces are order complete for
.
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