Band (order theory)

For any subset $S$ of a vector lattice $X,$ the set $S^\{\backslash perp\}$ of all elements of $X$ disjoint from $S$ is a band in $X.${{sfn|Narici|Beckenstein|2011|pp=204–214}}

If $\backslash mathcal\{L\}^p(\backslash mu)$ ($1\; \backslash leq\; p\; \backslash leq\; \backslash infty$) is the usual space of real valued functions used to define Lp spaces $L^p,$ then $\backslash mathcal\{L\}^p(\backslash mu)$ is countably order complete (that is, each subset that is bounded above has a supremum) but in general is not order complete.

If $N$ is the vector subspace of all $\backslash mu$-null functions then $N$ is a solid subset of $\backslash mathcal\{L\}^p(\backslash mu)$ that is {{em|not}} a band.{{sfn|Narici|Beckenstein|2011|pp=204–214}}

The intersection of an arbitrary family of bands in a vector lattice $X$ is a band in $X.${{sfn | Schaefer |Wolff| 1999 | pp=204–214}}

• {{annotated link|Solid set}}
• {{annotated link|Locally convex vector lattice}}
• {{annotated link|Vector lattice}}

{{reflist|group=note}}

{{reflist}}

• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Ordered topological vector spaces}}

{{Functional analysis}}

Category:Functional analysis