Solid set explained

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| \leq |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.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).