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 $x\in M.$Template:Sfn The smallest band containing a subset $S$ of $X$ is called the band generated by $S$ in $X.$Template:Sfn 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 $X.$Template:Sfn

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 Template:Em a band.Template:Sfn

Properties

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

See also

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References

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces