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 $S\subseteq M$ such that $x=\sup S$ exists in $X,$ we have $x\in M.$ The smallest band containing a subset $S$ of $X$ is called the band generated by $S$ in $X.$ A band generated by a singleton set is called a principal band.

Examples

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

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

Properties

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

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.