Band (order theory)

In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X is a subspace M of X that is solid and such that for all S ⊆ M such that x = sup S exists in X, we have x ∈ M.[1] The smallest band containing a subset S of X is called the band generated by S in X.[1] A band generated by a singleton set is called a principal band.

Examples

For any subset S of a vector lattice X, the set $S^{\perp }$ of all elements of X disjoint from S is a band in X.[1]

If ${\mathcal {L}}^{p}\left(\mu \right)$ ( $1\leq p\leq \infty$ ) is the usual space of real valued functions used to define L^ps, then ${\mathcal {L}}^{p}\left(\mu \right)$ is countably order complete (i.e. each subset that is bounded above has a supremum) but in general is not order complete. If N is the vector subspace of all $\mu$ -null functions then N is a solid subset of ${\mathcal {L}}^{p}\left(\mu \right)$ that is not a band.[1]

Properties

The intersection of an arbitrary family of bands in a vector lattice X is a band in X.[1]

References

Schaefer 1999, pp. 204–214.

Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)

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[FOOTNOTESchaefer1999204–214-1] Schaefer 1999, pp. 204–214.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Ordered topological vector spaces
Basic concepts	Ordered topological vector space Ordered vector space Partially ordered space Riesz space Order topology Order unit Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive
Main results	Freudenthal spectral

Band (order theory)

Examples

Properties

See also

References