Английская Википедия:Band (order theory)

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In mathematics, specifically in order theory and functional analysis, a band in a vector lattice <math>X</math> is a subspace <math>M</math> of <math>X</math> that is solid and such that for all <math>S \subseteq M</math> such that <math>x = \sup S</math> exists in <math>X,</math> we have <math>x \in M.</math>Шаблон:Sfn The smallest band containing a subset <math>S</math> of <math>X</math> is called the band generated by <math>S</math> in <math>X.</math>Шаблон:Sfn A band generated by a singleton set is called a principal band.

Examples

For any subset <math>S</math> of a vector lattice <math>X,</math> the set <math>S^{\perp}</math> of all elements of <math>X</math> disjoint from <math>S</math> is a band in <math>X.</math>Шаблон:Sfn

If <math>\mathcal{L}^p(\mu)</math> (<math>1 \leq p \leq \infty</math>) is the usual space of real valued functions used to define Lp spaces <math>L^p,</math> then <math>\mathcal{L}^p(\mu)</math> is countably order complete (that is, each subset that is bounded above has a supremum) but in general is not order complete. If <math>N</math> is the vector subspace of all <math>\mu</math>-null functions then <math>N</math> is a solid subset of <math>\mathcal{L}^p(\mu)</math> that is Шаблон:Em a band.Шаблон:Sfn

Properties

The intersection of an arbitrary family of bands in a vector lattice <math>X</math> is a band in <math>X.</math>Шаблон:Sfn

See also

• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link

References

• Шаблон:Narici Beckenstein Topological Vector Spaces
• Шаблон:Schaefer Wolff Topological Vector Spaces