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.
For any subset of a vector lattice the set of all elements of disjoint from is a band in
If () is the usual space of real valued functions used to define Lp spaces then is countably order complete (that is, each subset that is bounded above has a supremum) 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.
The intersection of an arbitrary family of bands in a vector lattice is a band in