In mathematics, specifically in

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, a subset

S

of a

vector lattice In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur ...

is said to be solid and is called an ideal if for all

s \in S

and

x \in X,

, x,  \leq , s,

then

x \in S.

ordered vector space In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Definition Given a vector space ''X'' over the real numbers R and a pr ...

whose order is Archimedean is said to be ''

Archimedean order In mathematics, specifically in order theory, a binary relation \,\leq\, on a vector space X over the real or complex numbers is called Archimedean if for all x \in X, whenever there exists some y \in X such that n x \leq y for all positive intege ...

ed''. 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. In a locally convex vector lattice

X,

the

polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...

of every solid neighborhood of the origin is a solid subset of the continuous dual space

X^

; moreover, the family of all solid equicontinuous subsets of

X^

is a fundamental family of equicontinuous sets, the polars (in bidual

X^

) form a neighborhood base of the origin for the natural topology on

X^

(that is, the topology of uniform convergence on equicontinuous subset of

X^

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).

References

* * {{Ordered topological vector spaces Functional analysis Order theory

Examples

Properties

See also

References