Solid set

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

An ordered vector space whose order is Archimedean is said to be Archimedean ordered.

An ideal generated by a singleton set is called a principal ideal in

The intersection of an arbitrary collection of ideals in

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