Vector bornology

In mathematics, especially functional analysis, a bornology [math]\displaystyle{ \mathcal{B} }[/math] on a vector space [math]\displaystyle{ X }[/math] over a field [math]\displaystyle{ \mathbb{K}, }[/math] where [math]\displaystyle{ \mathbb{K} }[/math] has a bornology ℬ_{[math]\displaystyle{ \mathbb{F} }[/math]}, is called a vector bornology if [math]\displaystyle{ \mathcal{B} }[/math] makes the vector space operations into bounded maps.

Definitions

Prerequisits

A bornology on a set [math]\displaystyle{ X }[/math] is a collection [math]\displaystyle{ \mathcal{B} }[/math] of subsets of [math]\displaystyle{ X }[/math] that satisfy all the following conditions:

1. [math]\displaystyle{ \mathcal{B} }[/math] covers [math]\displaystyle{ X; }[/math] that is, [math]\displaystyle{ X = \cup \mathcal{B} }[/math]
2. [math]\displaystyle{ \mathcal{B} }[/math] is stable under inclusions; that is, if [math]\displaystyle{ B \in \mathcal{B} }[/math] and [math]\displaystyle{ A \subseteq B, }[/math] then [math]\displaystyle{ A \in \mathcal{B} }[/math]
3. [math]\displaystyle{ \mathcal{B} }[/math] is stable under finite unions; that is, if [math]\displaystyle{ B_1, \ldots, B_n \in \mathcal{B} }[/math] then [math]\displaystyle{ B_1 \cup \cdots \cup B_n \in \mathcal{B} }[/math]

Elements of the collection [math]\displaystyle{ \mathcal{B} }[/math] are called [math]\displaystyle{ \mathcal{B} }[/math]-bounded or simply bounded sets if [math]\displaystyle{ \mathcal{B} }[/math] is understood. The pair [math]\displaystyle{ (X, \mathcal{B}) }[/math] is called a bounded structure or a bornological set.

A base or fundamental system of a bornology [math]\displaystyle{ \mathcal{B} }[/math] is a subset [math]\displaystyle{ \mathcal{B}_0 }[/math] of [math]\displaystyle{ \mathcal{B} }[/math] such that each element of [math]\displaystyle{ \mathcal{B} }[/math] is a subset of some element of [math]\displaystyle{ \mathcal{B}_0. }[/math] Given a collection [math]\displaystyle{ \mathcal{S} }[/math] of subsets of [math]\displaystyle{ X, }[/math] the smallest bornology containing [math]\displaystyle{ \mathcal{S} }[/math] is called the bornology generated by [math]\displaystyle{ \mathcal{S}. }[/math]

If [math]\displaystyle{ (X, \mathcal{B}) }[/math] and [math]\displaystyle{ (Y, \mathcal{C}) }[/math] are bornological sets then their product bornology on [math]\displaystyle{ X \times Y }[/math] is the bornology having as a base the collection of all sets of the form [math]\displaystyle{ B \times C, }[/math] where [math]\displaystyle{ B \in \mathcal{B} }[/math] and [math]\displaystyle{ C \in \mathcal{C}. }[/math] A subset of [math]\displaystyle{ X \times Y }[/math] is bounded in the product bornology if and only if its image under the canonical projections onto [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are both bounded.

If [math]\displaystyle{ (X, \mathcal{B}) }[/math] and [math]\displaystyle{ (Y, \mathcal{C}) }[/math] are bornological sets then a function [math]\displaystyle{ f : X \to Y }[/math] is said to be a locally bounded map or a bounded map (with respect to these bornologies) if it maps [math]\displaystyle{ \mathcal{B} }[/math]-bounded subsets of [math]\displaystyle{ X }[/math] to [math]\displaystyle{ \mathcal{C} }[/math]-bounded subsets of [math]\displaystyle{ Y; }[/math] that is, if [math]\displaystyle{ f\left(\mathcal{B}\right) \subseteq \mathcal{C}. }[/math] If in addition [math]\displaystyle{ f }[/math] is a bijection and [math]\displaystyle{ f^{-1} }[/math] is also bounded then [math]\displaystyle{ f }[/math] is called a bornological isomorphism.

Vector bornology

Let [math]\displaystyle{ X }[/math] be a vector space over a field [math]\displaystyle{ \mathbb{K} }[/math] where [math]\displaystyle{ \mathbb{K} }[/math] has a bornology [math]\displaystyle{ \mathcal{B}_{\mathbb{K}}. }[/math] A bornology [math]\displaystyle{ \mathcal{B} }[/math] on [math]\displaystyle{ X }[/math] is called a vector bornology on [math]\displaystyle{ X }[/math] if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).

If [math]\displaystyle{ X }[/math] is a vector space and [math]\displaystyle{ \mathcal{B} }[/math] is a bornology on [math]\displaystyle{ X, }[/math] then the following are equivalent:

1. [math]\displaystyle{ \mathcal{B} }[/math] is a vector bornology
2. Finite sums and balanced hulls of [math]\displaystyle{ \mathcal{B} }[/math]-bounded sets are [math]\displaystyle{ \mathcal{B} }[/math]-bounded
3. The scalar multiplication map [math]\displaystyle{ \mathbb{K} \times X \to X }[/math] defined by [math]\displaystyle{ (s, x) \mapsto sx }[/math] and the addition map [math]\displaystyle{ X \times X \to X }[/math] defined by [math]\displaystyle{ (x, y) \mapsto x + y, }[/math] are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets)

A vector bornology [math]\displaystyle{ \mathcal{B} }[/math] is called a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then [math]\displaystyle{ \mathcal{B}. }[/math] And a vector bornology [math]\displaystyle{ \mathcal{B} }[/math] is called separated if the only bounded vector subspace of [math]\displaystyle{ X }[/math] is the 0-dimensional trivial space [math]\displaystyle{ \{ 0 \}. }[/math]

Usually, [math]\displaystyle{ \mathbb{K} }[/math] is either the real or complex numbers, in which case a vector bornology [math]\displaystyle{ \mathcal{B} }[/math] on [math]\displaystyle{ X }[/math] will be called a convex vector bornology if [math]\displaystyle{ \mathcal{B} }[/math] has a base consisting of convex sets.

Characterizations

Suppose that [math]\displaystyle{ X }[/math] is a vector space over the field [math]\displaystyle{ \mathbb{F} }[/math] of real or complex numbers and [math]\displaystyle{ \mathcal{B} }[/math] is a bornology on [math]\displaystyle{ X. }[/math] Then the following are equivalent:

1. [math]\displaystyle{ \mathcal{B} }[/math] is a vector bornology
2. addition and scalar multiplication are bounded maps
3. the balanced hull of every element of [math]\displaystyle{ \mathcal{B} }[/math] is an element of [math]\displaystyle{ \mathcal{B} }[/math] and the sum of any two elements of [math]\displaystyle{ \mathcal{B} }[/math] is again an element of [math]\displaystyle{ \mathcal{B} }[/math]

Bornology on a topological vector space

If [math]\displaystyle{ X }[/math] is a topological vector space then the set of all bounded subsets of [math]\displaystyle{ X }[/math] from a vector bornology on [math]\displaystyle{ X }[/math] called the von Neumann bornology of [math]\displaystyle{ X }[/math], the usual bornology, or simply the bornology of [math]\displaystyle{ X }[/math] and is referred to as natural boundedness. In any locally convex topological vector space [math]\displaystyle{ X, }[/math] the set of all closed bounded disks form a base for the usual bornology of [math]\displaystyle{ X. }[/math]

Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.

Topology induced by a vector bornology

Suppose that [math]\displaystyle{ X }[/math] is a vector space over the field [math]\displaystyle{ \mathbb{K} }[/math] of real or complex numbers and [math]\displaystyle{ \mathcal{B} }[/math] is a vector bornology on [math]\displaystyle{ X. }[/math] Let [math]\displaystyle{ \mathcal{N} }[/math] denote all those subsets [math]\displaystyle{ N }[/math] of [math]\displaystyle{ X }[/math] that are convex, balanced, and bornivorous. Then [math]\displaystyle{ \mathcal{N} }[/math] forms a neighborhood basis at the origin for a locally convex topological vector space topology.

Examples

Locally convex space of bounded functions

Let [math]\displaystyle{ \mathbb{K} }[/math] be the real or complex numbers (endowed with their usual bornologies), let [math]\displaystyle{ (T, \mathcal{B}) }[/math] be a bounded structure, and let [math]\displaystyle{ LB(T, \mathbb{K}) }[/math] denote the vector space of all locally bounded [math]\displaystyle{ \mathbb{K} }[/math]-valued maps on [math]\displaystyle{ T. }[/math] For every [math]\displaystyle{ B \in \mathcal{B}, }[/math] let [math]\displaystyle{ p_{B}(f) := \sup \left| f(B) \right| }[/math] for all [math]\displaystyle{ f \in LB(T, \mathbb{K}), }[/math] where this defines a seminorm on [math]\displaystyle{ X. }[/math] The locally convex topological vector space topology on [math]\displaystyle{ LB(T, \mathbb{K}) }[/math] defined by the family of seminorms [math]\displaystyle{ \left\{ p_{B} : B \in \mathcal{B} \right\} }[/math] is called the topology of uniform convergence on bounded set. This topology makes [math]\displaystyle{ LB(T, \mathbb{K}) }[/math] into a complete space.

Bornology of equicontinuity

Let [math]\displaystyle{ T }[/math] be a topological space, [math]\displaystyle{ \mathbb{K} }[/math] be the real or complex numbers, and let [math]\displaystyle{ C(T, \mathbb{K}) }[/math] denote the vector space of all continuous [math]\displaystyle{ \mathbb{K} }[/math]-valued maps on [math]\displaystyle{ T. }[/math] The set of all equicontinuous subsets of [math]\displaystyle{ C(T, \mathbb{K}) }[/math] forms a vector bornology on [math]\displaystyle{ C(T, \mathbb{K}). }[/math]

See also

• Bornivorous set
• Bornological space
• Bornology
• Space of linear maps
• Ultrabornological space

Bibliography

• Template:Hogbe-Nlend Bornologies and Functional Analysis
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 978-082180780-4. 
• 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.