Band (order theory)

Not to be confused with Band (algebra).

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 ⊆ M such that x = sup S exists in X, we have x ∈ 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 ${\mathcal {L}}^{p}\left(\mu \right)$ ($1\leq p\leq \infty$) is the usual space of real valued functions used to define L^ps, then ${\mathcal {L}}^{p}\left(\mu \right)$ is countably order complete (i.e. 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 ${\mathcal {L}}^{p}\left(\mu \right)$ 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
• Vector lattice

References

• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)