Band (order theory)
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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
- Solid set
- Locally convex vector lattice
- Vector lattice – Partially ordered vector space, ordered as a lattice
References
- ^ a b c d Narici & Beckenstein 2011, pp. 204–214.
- ^ Schaefer & Wolff 1999, pp. 204–214.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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- Functional analysis