Band (order theory)
In mathematics, specifically in order theory and functional analysis, a band in a vector lattice is a subspace of that is solid and such that for all such that exists in we have
The smallest band containing a subset of is called the band generated by in
A band generated by a singleton set is called a principal band.
Examples
For any subset of a vector lattice the set of all elements of disjoint from is a band inIf is the usual space of real valued functions used to define Lp spaces then is countably order complete but in general is not order complete.
If is the vector subspace of all -null functions then is a solid subset of that is a band.