Topological Vector Lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) $X$ that has a partial order $\,\leq\,$ making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.
Ordered vector lattices have important applications in spectral theory.

Definition

If $X$ is a vector lattice then by the vector lattice operations we mean the following maps:
# the three maps $X$ to itself defined by $x \mapsto, x ,$, $x \mapsto x^+$, $x \mapsto x^$, and
# the two maps from $X \times X$ into $X$ defined by $(x, y) \mapsto \sup_ \$ and$(x, y) \mapsto \inf_ \$.
If $X$ is a TVS over the reals and a vector lattice, then $X$ is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.

If $X$ 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.

If $X$ is a topological vector space (TVS) and an ordered vector space then $X$ is called locally solid if $X$ possesses a neighborhood base at the origin consisting of solid sets.
A topological vector lattice is a Hausdorff TVS $X$ that has a partial order $\,\leq\,$ making it into vector lattice that is locally solid.

Properties

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.
Let $\mathcal$ denote the set of all bounded subsets of a topological vector lattice with positive cone $C$ and for any subset $S$, let C := (S + C) \cap (S - C) be the $C$-saturated hull of $S$.
Then the topological vector lattice's positive cone $C$ is a strict $\mathcal$-cone, where $C$ is a strict $\mathcal$-cone means that $\left\$ is a fundamental subfamily of $\mathcal$ that is, every $B \in \mathcal$ is contained as a subset of some element of $\left\$).

If a topological vector lattice $X$ is order complete then every band is closed in $X$.

Examples

The Banach spaces $L^p(\mu)$ ($1 \leq p \leq \infty$) are Banach lattices under their canonical orderings.
These spaces are order complete for $p < \infty$.

See also

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References

Bibliography

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{{Order theory

Functional analysis