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Concept
Solid set
Summary
In mathematics, specifically in order theory and functional analysis, a subset of a vector lattice is said to be solid and is called an ideal if for all and if then
An ordered vector space whose order is Archimedean is said to be Archimedean ordered.
If then the ideal generated by is the smallest ideal in containing
An ideal generated by a singleton set is called a principal ideal in
The intersection of an arbitrary collection of ideals in is again an ideal and furthermore, is clearly an ideal of itself;
thus every subset of is contained in a unique smallest ideal.
In a locally convex vector lattice the polar of every solid neighborhood of the origin is a solid subset of the continuous dual space ;
moreover, the family of all solid equicontinuous subsets of is a fundamental family of equicontinuous sets, the polars (in bidual ) form a neighborhood base of the origin for the natural topology on (that is, the topology of uniform convergence on equicontinuous subset of ).
A solid subspace of a vector lattice is necessarily a sublattice of
If is a solid subspace of a vector lattice then the quotient is a vector lattice (under the canonical order).
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