Topological vector lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order ≤ making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.[1] Ordered vector lattices have important applications in spectral theory.

## Contents

* 1 Definition
* 2 Properties
* 3 Examples
* 4 See also
* 5 References
* 6 Bibliography

## Definition

If X is a vector lattice then by the vector lattice operations we mean the following maps:

1. the three maps X to itself defined by x↦|x|, x↦x+, x↦x−, and
2. the two maps from X×X into X defined by (x,y)↦sup{x,y} and(x,y)↦inf{x,y}.

If X is a TVS over the reals and a vector lattice, then X is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.[1]

If X is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous.[1]

If X is a topological vector space (TVS) and an ordered vector space then X is called locally solid if X possesses a neighborhood base at the origin consisting of solid sets.[1] A topological vector lattice is a Hausdorff TVS X that has a partial order ≤ making it into vector lattice that is locally solid.[1]

## Properties

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.[1] Let ℬ denote the set of all bounded subsets of a topological vector lattice with positive cone C and for any subset S, let [S]C:=(S+C)∩(S−C) be the C-saturated hull of S. Then the topological vector lattice's positive cone C is a strict ℬ-cone,[1] where C is a strict ℬ-cone means that {[B]C:B∈ℬ} is a fundamental subfamily of ℬ that is, every B∈ℬ is contained as a subset of some element of {[B]C:B∈ℬ}).[2]

If a topological vector lattice X is order complete then every band is closed in X.[1]

## Examples

The Banach spaces Lp(μ) (1≤p≤∞) are Banach lattices under their canonical orderings. These spaces are order complete for p<∞.

## See also

* Banach lattice - Banach space with a compatible structure of a lattice
* Complemented lattice - Bound lattice in which every element has a complement
* Fréchet lattice - Topological vector lattice
* Locally convex vector lattice
* Ordered vector space - Vector space with a partial order
* Pseudocomplement
* Riesz space - Partially ordered vector space, ordered as a lattice

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Schaefer & Wolff 1999, pp. 234–242.
2. ↑ Schaefer & Wolff 1999, pp. 215–222.

## Bibliography

* 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

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