topological vector lattice

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 $x\; \backslash mapsto|x\; |$, $x\; \backslash mapsto\; x^+$, $x\; \backslash mapsto\; x^\{-\}$, and
2. the two maps from $X\; \backslash times\; X$ into $X$ defined by $(x,\; y)\; \backslash mapsto\; \backslash sup\_\{\}\; \backslash \{\; x,\; y\; \backslash \}$ and$(x,\; y)\; \backslash mapsto\; \backslash inf\_\{\}\; \backslash \{\; x,\; y\; \backslash \}$.

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

A topological vector lattice is a Hausdorff TVS $X$ that has a partial order $\backslash ,\backslash leq\backslash ,$ making it into vector lattice that is locally solid.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

Let $\backslash mathcal\{B\}$ denote the set of all bounded subsets of a topological vector lattice with positive cone $C$ and for any subset $S$, let $[S]\_C\; :=\; (S\; +\; C)\; \backslash cap\; (S\; -\; C)$ be the $C$-saturated hull of $S$.

Then the topological vector lattice's positive cone $C$ is a strict $\backslash mathcal\{B\}$-cone,{{sfn|Schaefer|Wolff|1999|pp=234–242}} where $C$ is a strict $\backslash mathcal\{B\}$-cone means that $\backslash left\backslash \{\; [B]\_C\; :\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{B\}$ that is, every $B\; \backslash in\; \backslash mathcal\{B\}$ is contained as a subset of some element of $\backslash left\backslash \{\; [B]\_C\; :\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash right\backslash \}$).{{sfn|Schaefer|Wolff|1999|pp=215–222}}

If a topological vector lattice $X$ is order complete then every band is closed in $X$.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

The L^p spaces ($1\; \backslash leq\; p\; \backslash leq\; \backslash infty$) are Banach lattices under their canonical orderings.

These spaces are order complete for .

• {{annotated link|Banach lattice}}
• {{annotated link|Complemented lattice}}
• {{annotated link|Fréchet lattice}}
• {{annotated link|Locally convex vector lattice}}
• {{annotated link|Normed lattice}}
• {{annotated link|Ordered vector space}}
• {{annotated link|Pseudocomplement}}
• {{annotated link|Riesz space}}

{{reflist|group=note}}

{{reflist}}

• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Functional analysis}}

{{Ordered topological vector spaces}}

{{Order theory}}

Category:Functional analysis