Vector bornology

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

Definitions

= Prerequisits =

A {{em|bornology}} on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ that satisfy all the following conditions:

$\mathcal{B}$ covers $X;$ that is, $X = \cup \mathcal{B}$
$\mathcal{B}$ is stable under inclusions; that is, if $B \in \mathcal{B}$ and $A \subseteq B,$ then $A \in \mathcal{B}$
$\mathcal{B}$ is stable under finite unions; that is, if $B_1, \ldots, B_n \in \mathcal{B}$ then $B_1 \cup \cdots \cup B_n \in \mathcal{B}$

Elements of the collection $\mathcal{B}$ are called {{em| $\mathcal{B}$ -bounded}} or simply {{em|bounded sets}} if $\mathcal{B}$ is understood.

The pair $(X, \mathcal{B})$ is called a {{em|bounded structure}} or a {{em|bornological set}}.

A {{em|base}} or {{em|fundamental system}} of a bornology $\mathcal{B}$ is a subset $\mathcal{B}_0$ of $\mathcal{B}$ such that each element of $\mathcal{B}$ is a subset of some element of $\mathcal{B}_0.$ Given a collection $\mathcal{S}$ of subsets of $X,$ the smallest bornology containing $\mathcal{S}$ is called the bornology generated by $\mathcal{S}.$ {{sfn|Narici|Beckenstein|2011|pp=156–175}}

If $(X, \mathcal{B})$ and $(Y, \mathcal{C})$ are bornological sets then their {{em|product bornology}} on $X \times Y$ is the bornology having as a base the collection of all sets of the form $B \times C,$ where $B \in \mathcal{B}$ and $C \in \mathcal{C}.$ {{sfn|Narici|Beckenstein|2011|pp=156–175}}

A subset of $X \times Y$ is bounded in the product bornology if and only if its image under the canonical projections onto $X$ and $Y$ are both bounded.

If $(X, \mathcal{B})$ and $(Y, \mathcal{C})$ are bornological sets then a function $f : X \to Y$ is said to be a {{em|locally bounded map}} or a {{em|bounded map}} (with respect to these bornologies) if it maps $\mathcal{B}$ -bounded subsets of $X$ to $\mathcal{C}$ -bounded subsets of $Y;$ that is, if $f\left(\mathcal{B}\right) \subseteq \mathcal{C}.$ {{sfn|Narici|Beckenstein|2011|pp=156–175}}

If in addition $f$ is a bijection and $f^{-1}$ is also bounded then $f$ is called a {{em|bornological isomorphism}}.

= Vector bornology =

Let $X$ be a vector space over a field $\mathbb{K}$ where $\mathbb{K}$ has a bornology $\mathcal{B}_{\mathbb{K}}.$

A bornology $\mathcal{B}$ on $X$ is called a {{em|vector bornology on $X$ }} 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 $X$ is a vector space and $\mathcal{B}$ is a bornology on $X,$ then the following are equivalent:

$\mathcal{B}$ is a vector bornology
Finite sums and balanced hulls of $\mathcal{B}$ -bounded sets are $\mathcal{B}$ -bounded{{sfn|Narici|Beckenstein|2011|pp=156-175}}
The scalar multiplication map $\mathbb{K} \times X \to X$ defined by $(s, x) \mapsto sx$ and the addition map $X \times X \to X$ defined by $(x, y) \mapsto x + y,$ are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets){{sfn|Narici|Beckenstein|2011|pp=156-175}}

A vector bornology $\mathcal{B}$ is called a {{em|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 $\mathcal{B}.$

And a vector bornology $\mathcal{B}$ is called {{em|separated}} if the only bounded vector subspace of $X$ is the 0-dimensional trivial space $\{ 0 \}.$

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

Characterizations

Suppose that $X$ is a vector space over the field $\mathbb{F}$ of real or complex numbers and $\mathcal{B}$ is a bornology on $X.$

Then the following are equivalent:

$\mathcal{B}$ is a vector bornology
addition and scalar multiplication are bounded maps{{sfn|Narici|Beckenstein|2011|pp=156–175}}
the balanced hull of every element of $\mathcal{B}$ is an element of $\mathcal{B}$ and the sum of any two elements of $\mathcal{B}$ is again an element of $\mathcal{B}$ {{sfn|Narici|Beckenstein|2011|pp=156–175}}

Bornology on a topological vector space

If $X$ is a topological vector space then the set of all bounded subsets of $X$ from a vector bornology on $X$ called the {{em|von Neumann bornology of $X$ }}, the {{em|usual bornology}}, or simply the {{em|bornology}} of $X$ and is referred to as {{em|natural boundedness}}.{{sfn|Narici|Beckenstein|2011|pp=156–175}}

In any locally convex topological vector space $X,$ the set of all closed bounded disks form a base for the usual bornology of $X.$ {{sfn|Narici|Beckenstein|2011|pp=156–175}}

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 $X$ is a vector space over the field $\mathbb{K}$ of real or complex numbers and $\mathcal{B}$ is a vector bornology on $X.$

Let $\mathcal{N}$ denote all those subsets $N$ of $X$ that are convex, balanced, and bornivorous.

Then $\mathcal{N}$ forms a neighborhood basis at the origin for a locally convex topological vector space topology.

Examples

= Locally convex space of bounded functions =

Let $\mathbb{K}$ be the real or complex numbers (endowed with their usual bornologies), let $(T, \mathcal{B})$ be a bounded structure, and let $LB(T, \mathbb{K})$ denote the vector space of all locally bounded $\mathbb{K}$ -valued maps on $T.$

For every $B \in \mathcal{B},$ let $p_{B}(f) := \sup \left| f(B) \right|$ for all $f \in LB(T, \mathbb{K}),$ where this defines a seminorm on $X.$

The locally convex topological vector space topology on $LB(T, \mathbb{K})$ defined by the family of seminorms $\left\{ p_{B} : B \in \mathcal{B} \right\}$ is called the {{em|topology of uniform convergence on bounded set}}.{{sfn|Narici|Beckenstein|2011|pp=156–175}}

This topology makes $LB(T, \mathbb{K})$ into a complete space.{{sfn|Narici|Beckenstein|2011|pp=156–175}}

= Bornology of equicontinuity =

Let $T$ be a topological space, $\mathbb{K}$ be the real or complex numbers, and let $C(T, \mathbb{K})$ denote the vector space of all continuous $\mathbb{K}$ -valued maps on $T.$

The set of all equicontinuous subsets of $C(T, \mathbb{K})$ forms a vector bornology on $C(T, \mathbb{K}).$ {{sfn|Narici|Beckenstein|2011|pp=156–175}}

Citations

Bibliography

{{Hogbe-Nlend Bornologies and Functional Analysis}}
{{Cite book| title = The Convenient Setting of Global Analysis

| last1 = Kriegl | first1 = Andreas

| last2 = Michor | first2 = Peter W.

| year = 1997

| publisher = American Mathematical Society

| series = Mathematical Surveys and Monographs

| isbn = 978-082180780-4

}}

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

Category:Topological vector spaces