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 ≤ 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:

1. the three maps X to itself defined by $x\mapsto |x|$, $x\mapsto x^{+}$, $x\mapsto x^{-}$, and
2. the two maps from $X\times X$ into X defined by $(x,y)\mapsto \sup \left\{x,y\right\}$ and$(x,y)\mapsto \inf \left\{x,y\right\}$.

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 ≤ 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 ${\mathcal {B}}$ denote the set of all bounded subsets of a topological vector lattice with positive cone C and for any subset S, let $[S]_{C}:=\left(S+C\right)\cap \left(S-C\right)$ be the C-saturated hull of S. Then the topological vector lattice's positive cone C is a strict ${\mathcal {B}}$-cone, where C is a strict ${\mathcal {B}}$-cone means that $\left\{\left[B\right]_{C}:B\in {\mathcal {B}}\right\}$ is a fundamental subfamily of ${\mathcal {B}}$ (i.e. every $B\in {\mathcal {B}}$ is contained as a subset of some element of $\left\{\left[B\right]_{C}:B\in {\mathcal {B}}\right\}$).

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

Examples

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

See also

• Banach lattice
• Fréchet lattice
• Locally convex vector lattice
• Normed lattice
• Ordered vector space
• Riesz space

References

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.