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| ≤ |s| then x belongs to S. An ordered vector space whose order is Archimedean is said to be Archimedean ordered.[1] If S is a subset of 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 0 is a solid subset of the continuous dual space ${\displaystyle X^{\prime }}$; moreover, the family of all solid equicontinuous subsets of ${\displaystyle X^{\prime }}$ is a fundamental family of equicontinuous sets, the polars (in bidual ${\displaystyle X^{\prime \prime }}$) form a neighborhood base of the origin for the natural topology on ${\displaystyle X^{\prime \prime }}$ (i.e. the topology of uniform convergence on equicontinuous subset of ${\displaystyle X^{\prime }}$).[2]

Properties

• A solid subspace of a vector lattice X is necessarily a sublattice of X.[1]
• If N is a solid subspace of a vector lattice X then the quotient X/N is a vector lattice (under the canonical order).[1]

See also

• Vector lattice

References

1. Schaefer & Wolff 1999, pp. 204–214.
2. Schaefer & Wolff 1999, pp. 234–242.

• 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.