Vector bornology

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

Definitions

Prerequisits

A Template:Em on a set $X$ is a collection $ℬ$ of subsets of $X$ that satisfy all the following conditions:

1. $ℬ$ covers $X;$ that is, $X=\cup ℬ$
2. $ℬ$ is stable under inclusions; that is, if $B\in ℬ$ and $A\subseteq B,$ then $A\in ℬ$
3. $ℬ$ is stable under finite unions; that is, if $B_1,\dots ,B_n\in ℬ$ then $B_1\cup \cdots \cup B_n\in ℬ$

Elements of the collection $ℬ$ are called Template:Em or simply Template:Em if $ℬ$ is understood. The pair $(X,ℬ)$ is called a Template:Em or a Template:Em.

A Template:Em or Template:Em of a bornology $ℬ$ is a subset ${ℬ}_0$ of $ℬ$ such that each element of $ℬ$ is a subset of some element of ${ℬ}_0.$ Given a collection $𝒮$ of subsets of $X,$ the smallest bornology containing $𝒮$ is called the bornology generated by $𝒮.$Template:Sfn

If $(X,ℬ)$ and $(Y,𝒞)$ are bornological sets then their Template:Em on $X\times Y$ is the bornology having as a base the collection of all sets of the form $B\times C,$ where $B\in ℬ$ and $C\in 𝒞.$Template:Sfn 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 $(X,ℬ)$ and $(Y,𝒞)$ are bornological sets then a function $f:X\to Y$ is said to be a Template:Em or a Template:Em (with respect to these bornologies) if it maps $ℬ$-bounded subsets of $X$ to $𝒞$-bounded subsets of $Y;$ that is, if $f\left(ℬ\right)\subseteq 𝒞.$Template:Sfn If in addition $f$ is a bijection and $f^{-1}$ is also bounded then $f$ is called a Template:Em.

Vector bornology

Let $X$ be a vector space over a field $𝕂$ where $𝕂$ has a bornology ${ℬ}_{𝕂}.$ A bornology $ℬ$ on $X$ is called a Template:Em 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 $ℬ$ is a bornology on $X,$ then the following are equivalent:

1. $ℬ$ is a vector bornology
2. Finite sums and balanced hulls of $ℬ$-bounded sets are $ℬ$-boundedTemplate:Sfn
3. The scalar multiplication map $𝕂\times X\to X$ 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)Template:Sfn

A vector bornology $ℬ$ is called a Template:Em if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then $ℬ.$ And a vector bornology $ℬ$ is called Template:Em if the only bounded vector subspace of $X$ is the 0-dimensional trivial space $\{0\}.$

Usually, $𝕂$ is either the real or complex numbers, in which case a vector bornology $ℬ$ on $X$ will be called a Template:Em if $ℬ$ has a base consisting of convex sets.

Characterizations

Suppose that $X$ is a vector space over the field $𝔽$ of real or complex numbers and $ℬ$ is a bornology on $X.$ Then the following are equivalent:

1. $ℬ$ is a vector bornology
2. addition and scalar multiplication are bounded mapsTemplate:Sfn
3. the balanced hull of every element of $ℬ$ is an element of $ℬ$ and the sum of any two elements of $ℬ$ is again an element of $ℬ$Template:Sfn

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 Template:Em, the Template:Em, or simply the Template:Em of $X$ and is referred to as Template:Em.Template:Sfn In any locally convex topological vector space $X,$ the set of all closed bounded disks form a base for the usual bornology of $X.$Template:Sfn

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 $𝕂$ of real or complex numbers and $ℬ$ is a vector bornology on $X.$ Let $𝒩$ denote all those subsets $N$ of $X$ that are convex, balanced, and bornivorous. Then $𝒩$ forms a neighborhood basis at the origin for a locally convex topological vector space topology.

Examples

Locally convex space of bounded functions

Let $𝕂$ be the real or complex numbers (endowed with their usual bornologies), let $(T,ℬ)$ be a bounded structure, and let $LB(T,𝕂)$ denote the vector space of all locally bounded $𝕂$-valued maps on $T.$ For every $B\in ℬ,$ let $p_B(f):=\sup |f(B)|$ for all $f\in LB(T,𝕂),$ where this defines a seminorm on $X.$ The locally convex topological vector space topology on $LB(T,𝕂)$ defined by the family of seminorms $\{p_B:B\in ℬ\}$ is called the Template:Em.Template:Sfn This topology makes $LB(T,𝕂)$ into a complete space.Template:Sfn

Bornology of equicontinuity

Let $T$ be a topological space, $𝕂$ be the real or complex numbers, and let $C(T,𝕂)$ denote the vector space of all continuous $𝕂$-valued maps on $T.$ The set of all equicontinuous subsets of $C(T,𝕂)$ forms a vector bornology on $C(T,𝕂).$Template:Sfn

See also

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

Bibliography

• Template:Hogbe-Nlend Bornologies and Functional Analysis
• Template:Cite book
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces