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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 [1] The smallest band containing a subset of is called the band generated by in [1] 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 in [1]

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 not a band.[1]

Properties

The intersection of an arbitrary family of bands in a vector lattice is a band in [2]

See also

References

  1. ^ a b c d Narici & Beckenstein 2011, pp. 204–214.
  2. ^ Schaefer & Wolff 1999, pp. 204–214.