In mathematics, specifically in order theory and functional analysis, a subset S {\displaystyle S} of a vector lattice X {\displaystyle X} is said to be solid and is called an ideal if for all s ∈ S {\displaystyle s\in S} and x ∈ X , {\displaystyle x\in X,} if | x | ≤ | s | {\displaystyle |x|\leq |s|} then x ∈ S . {\displaystyle x\in S.} An ordered vector space whose order is Archimedean is said to be Archimedean ordered. If S ⊆ X {\displaystyle S\subseteq X} then the ideal generated by S {\displaystyle S} is the smallest ideal in X {\displaystyle X} containing S . {\displaystyle S.} An ideal generated by a singleton set is called a principal ideal in X . {\displaystyle X.}
Examples The intersection of an arbitrary collection of ideals in X {\displaystyle X} is again an ideal and furthermore, X {\displaystyle X} is clearly an ideal of itself. thus every subset of X {\displaystyle X} is contained in a unique smallest ideal. In a locally convex vector lattice X , {\displaystyle X,} the polar of every solid neighborhood of the origin is a solid subset of the continuous dual space X ′ {\displaystyle X^{\prime }} ; moreover, the family of all solid equicontinuous subsets of X ′ {\displaystyle X^{\prime }} is a fundamental family of equicontinuous sets, the polars (in bidual X ′ ′ {\displaystyle X^{\prime \prime }} ) form a neighborhood base of the origin for the natural topology on X ′ ′ {\displaystyle X^{\prime \prime }} (that is, the topology of uniform convergence on equicontinuous subsets of X ′ {\displaystyle X^{\prime }} ).
Properties A solid subspace of a vector lattice X {\displaystyle X} is necessarily a sublattice of X . {\displaystyle X.}
If N {\displaystyle N} is a solid subspace of a vector lattice X {\displaystyle X} then the quotient X / N {\displaystyle X/N} is a vector lattice (under the canonical order).
See also Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets
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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.