Band (order theory)

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

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

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

The intersection of an arbitrary family of bands in a vector lattice ${\displaystyle X}$is a band in ${\displaystyle X.}$

• Solid set
• Locally convex vector lattice
• 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.