Band (order theory)

In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X is a subspace M of X that is solid and such that for all S ⊆ M such that x = sup S exists in X, we have x ∈ M.[1] The smallest band containing a subset S of X is called the band generated by S in X.[1] A band generated by a singleton set is called a principal band.

Examples

For any subset S of a vector lattice X, the set ${\displaystyle S^{\perp }}$ of all elements of X disjoint from S is a band in X.[1]

If ${\displaystyle {\mathcal {L}}^{p}\left(\mu \right)}$ (${\displaystyle 1\leq p\leq \infty }$) is the usual space of real valued functions used to define L^ps, then ${\displaystyle {\mathcal {L}}^{p}\left(\mu \right)}$ is countably order complete (i.e. each subset that is bounded above has a supremum) but in general is not order complete. If N is the vector subspace of all ${\displaystyle \mu }$-null functions then N is a solid subset of ${\displaystyle {\mathcal {L}}^{p}\left(\mu \right)}$ that is not a band.[1]

Properties

The intersection of an arbitrary family of bands in a vector lattice X is a band in X.[1]

See also

• Solid set
• Locally convex vector lattice
• Vector lattice

References

1. Schaefer 1999, pp. 204–214.

• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)