solid set

In mathematics, specifically in order theory and functional analysis, a subset $S$ of a vector lattice $X$ is said to be solid and is called an ideal if for all $s \in S$ and $x \in X,$ if $|x| \leq |s|$ then $x \in S.$

An ordered vector space whose order is Archimedean is said to be Archimedean ordered.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

If $S\subseteq X$ then the ideal generated by $S$ is the smallest ideal in $X$ containing $S.$

An ideal generated by a singleton set is called a principal ideal in $X.$

Examples

The intersection of an arbitrary collection of ideals in $X$ is again an ideal and furthermore, $X$ is clearly an ideal of itself;

thus every subset of $X$ is contained in a unique smallest ideal.

In a locally convex vector lattice $X,$ the polar of every solid neighborhood of the origin is a solid subset of the continuous dual space $X^{\prime}$ ;

moreover, the family of all solid equicontinuous subsets of $X^{\prime}$ is a fundamental family of equicontinuous sets, the polars (in bidual $X^{\prime\prime}$ ) form a neighborhood base of the origin for the natural topology on $X^{\prime\prime}$ (that is, the topology of uniform convergence on equicontinuous subset of $X^{\prime}$ ).{{sfn|Schaefer|Wolff|1999|pp=234–242}}

Properties

A solid subspace of a vector lattice $X$ is necessarily a sublattice of $X.$ {{sfn|Schaefer|Wolff|1999|pp=204–214}}
If $N$ is a solid subspace of a vector lattice $X$ then the quotient $X/N$ is a vector lattice (under the canonical order).{{sfn|Schaefer|Wolff|1999|pp=204–214}}

References

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

Category:Functional analysis

Category:Order theory

solid set

Examples

Properties

See also

References