Band (order theory)
In mathematics, specifically in order theory and functional analysis, a band in a vector lattice formula_1 is a subspace formula_2 of formula_1 that is solid and such that for all formula_4 such that formula_5 exists in formula_6 we have formula_7 The smallest band containing a subset formula_8 of formula_1 is called the band generated by formula_8 in formula_11 A band generated by a singleton set is called a principal band. Examples. For any subset formula_8 of a vector lattice formula_6 the set formula_14 of all elements of formula_1 disjoint from formula_8 is a band in formula_11 If  formula_18 (formula_19) is the usual space of real valued functions used to define Lp spaces formula_20 then formula_18 is countably order complete (that is, each subset that is bounded above has a supremum) but in general is not order complete. If formula_22 is the vector subspace of all formula_23-null functions then formula_22 is a solid subset of formula_18 that is a band. Properties. The intersection of an arbitrary family of bands in a vector lattice formula_1 is a band in formula_11