Solid set

In mathematics, specifically in order theory and functional analysis, a subset ${\displaystyle S}${\displaystyle S} of a vector lattice ${\displaystyle X}${\displaystyle X} is said to be solid and is called an ideal if for all ${\displaystyle s\in S}${\displaystyle s\in S} and ${\displaystyle x\in X,}${\displaystyle x\in X,} if ${\displaystyle |x|\leq |s|}${\displaystyle |x|\leq |s|} then ${\displaystyle x\in S.}${\displaystyle x\in S.} An ordered vector space whose order is Archimedean is said to be Archimedean ordered .^[1] If ${\displaystyle S\subseteq X}${\displaystyle S\subseteq X} then the ideal generated by ${\displaystyle S}${\displaystyle S} is the smallest ideal in ${\displaystyle X}${\displaystyle X} containing ${\displaystyle S.}${\displaystyle S.} An ideal generated by a singleton set is called a principal ideal in ${\displaystyle X.}${\displaystyle X.}

The intersection of an arbitrary collection of ideals in ${\displaystyle X}${\displaystyle X} is again an ideal and furthermore, ${\displaystyle X}${\displaystyle X} is clearly an ideal of itself; thus every subset of ${\displaystyle X}${\displaystyle X} is contained in a unique smallest ideal.

In a locally convex vector lattice ${\displaystyle X,}${\displaystyle X,} the polar of every solid neighborhood of the origin is a solid subset of the continuous dual space ${\displaystyle X^{\prime }}${\displaystyle X^{\prime }}; moreover, the family of all solid equicontinuous subsets of ${\displaystyle X^{\prime }}${\displaystyle X^{\prime }} is a fundamental family of equicontinuous sets, the polars (in bidual ${\displaystyle X^{\prime \prime }}${\displaystyle X^{\prime \prime }}) form a neighborhood base of the origin for the natural topology on ${\displaystyle X^{\prime \prime }}${\displaystyle X^{\prime \prime }} (that is, the topology of uniform convergence on equicontinuous subset of ${\displaystyle X^{\prime }}${\displaystyle X^{\prime }}).^[2]

• A solid subspace of a vector lattice ${\displaystyle X}${\displaystyle X} is necessarily a sublattice of ${\displaystyle X.}${\displaystyle X.}^[1]
• If ${\displaystyle N}${\displaystyle N} is a solid subspace of a vector lattice ${\displaystyle X}${\displaystyle X} then the quotient ${\displaystyle X/N}${\displaystyle X/N} is a vector lattice (under the canonical order).^[1]

• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

• 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.

• Functional analysis
• Order theory

• Pages displaying short descriptions of redirect targets via Module:Annotated link