Solid set

In mathematics, specifically in order theory and functional analysis, a subset $S$ of a vector lattice is said to be solid and is called an ideal if for all $s\in S$ and $x\in X,$ if $|x|\le |s|$ then $x\in S.$ An ordered vector space whose order is Archimedean is said to be Archimedean ordered. 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 $X^{\prime }$).

Properties

• A solid subspace of a vector lattice $X$ is necessarily a sublattice of $X.$
• If $N$ is a solid subspace of a vector lattice $X$ then the quotient $X/N$ is a vector lattice (under the canonical order).

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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.