Band (order theory)

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

Examples

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

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

Properties

The intersection of an arbitrary family of bands in a vector lattice [math]\displaystyle{ X }[/math] is a band in [math]\displaystyle{ X. }[/math]

See also

• Solid set
• Locally convex vector lattice

References

• 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.