Band (order theory)

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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 ${\displaystyle x\in M.}$^[1] The smallest band containing a subset ${\displaystyle S}$ of ${\displaystyle X}$ is called the band generated by ${\displaystyle S}$ in ${\displaystyle X.}$^[1] A band generated by a singleton set is called a principal band.

Examples[]

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 ${\displaystyle X.}$^[1]

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.^[1]

Properties[]

The intersection of an arbitrary family of bands in a vector lattice ${\displaystyle X}$ is a band in ${\displaystyle X.}$^[2]

See also[]

• Solid set
• Locally convex vector lattice
• 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.