Solid set

In mathematics, specifically in order theory and functional analysis, a subset [math]\displaystyle{ S }[/math] of a vector lattice is said to be solid and is called an ideal if for all [math]\displaystyle{ s \in S }[/math] and [math]\displaystyle{ x \in X, }[/math] if [math]\displaystyle{ |x| \leq |s| }[/math] then [math]\displaystyle{ x \in S. }[/math] An ordered vector space whose order is Archimedean is said to be Archimedean ordered. If [math]\displaystyle{ S\subseteq X }[/math] then the ideal generated by [math]\displaystyle{ S }[/math] is the smallest ideal in [math]\displaystyle{ X }[/math] containing [math]\displaystyle{ S. }[/math] An ideal generated by a singleton set is called a principal ideal in [math]\displaystyle{ X. }[/math]

Examples

The intersection of an arbitrary collection of ideals in [math]\displaystyle{ X }[/math] is again an ideal and furthermore, [math]\displaystyle{ X }[/math] is clearly an ideal of itself; thus every subset of [math]\displaystyle{ X }[/math] is contained in a unique smallest ideal.

In a locally convex vector lattice [math]\displaystyle{ X, }[/math] the polar of every solid neighborhood of the origin is a solid subset of the continuous dual space [math]\displaystyle{ X^{\prime} }[/math]; moreover, the family of all solid equicontinuous subsets of [math]\displaystyle{ X^{\prime} }[/math] is a fundamental family of equicontinuous sets, the polars (in bidual [math]\displaystyle{ X^{\prime\prime} }[/math]) form a neighborhood base of the origin for the natural topology on [math]\displaystyle{ X^{\prime\prime} }[/math] (that is, the topology of uniform convergence on equicontinuous subset of [math]\displaystyle{ X^{\prime} }[/math]).

Properties

• A solid subspace of a vector lattice [math]\displaystyle{ X }[/math] is necessarily a sublattice of [math]\displaystyle{ X. }[/math]
• If [math]\displaystyle{ N }[/math] is a solid subspace of a vector lattice [math]\displaystyle{ X }[/math] then the quotient [math]\displaystyle{ X/N }[/math] 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.