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 and if then An ordered vector space whose order is Archimedean is said to be Archimedean ordered.[1] 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

Examples

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 ).[2]

Properties

  • A solid subspace of a vector lattice is necessarily a sublattice of [1]
  • If is a solid subspace of a vector lattice then the quotient is a vector lattice (under the canonical order).[1]

See also

  • Vector lattice – Partially ordered vector space, ordered as a lattice

References

  1. ^ a b c Schaefer & Wolff 1999, pp. 204–214.
  2. ^ Schaefer & Wolff 1999, pp. 234–242.
  • 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.