Vector Bornology

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

Definitions

Prerequisits

A on a set $X$ is a collection $\mathcal$ of subsets of $X$ that satisfy all the following conditions:
#$\mathcal$ covers $X;$ that is, $X = \cup \mathcal$
#$\mathcal$ is stable under inclusions; that is, if $B \in \mathcal$ and $A \subseteq B,$ then $A \in \mathcal$
#$\mathcal$ is stable under finite unions; that is, if $B_1, \ldots, B_n \in \mathcal$ then $B_1 \cup \cdots \cup B_n \in \mathcal$

Elements of the collection $\mathcal$ are called or simply if $\mathcal$ is understood.
The pair $(X, \mathcal)$ is called a or a .

A or of a bornology $\mathcal$ is a subset $\mathcal_0$ of $\mathcal$ such that each element of $\mathcal$ is a subset of some element of $\mathcal_0.$ Given a collection $\mathcal$ of subsets of $X,$ the smallest bornology containing $\mathcal$ is called the bornology generated by $\mathcal.$

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

Vector bornology

Let $X$ be a vector space over a field $\mathbb$ where $\mathbb$ has a bornology $\mathcal_.$
A bornology $\mathcal$ on $X$ is called a 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 $\mathcal$ is a bornology on $X,$ then the following are equivalent:
#$\mathcal$ is a vector bornology
#Finite sums and balanced hulls of $\mathcal$-bounded sets are $\mathcal$-bounded
#The scalar multiplication map $\mathbb \times X \to X$ 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)

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

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

Characterizations

Suppose that $X$ is a vector space over the field $\mathbb$ of real or complex numbers and $\mathcal$ is a bornology on $X.$
Then the following are equivalent:
#$\mathcal$ is a vector bornology
#addition and scalar multiplication are bounded maps
#the balanced hull of every element of $\mathcal$ is an element of $\mathcal$ and the sum of any two elements of $\mathcal$ is again an element of $\mathcal$

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

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

Examples

Locally convex space of bounded functions

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

Bornology of equicontinuity

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

See also

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

Citations

Bibliography

*
*
*
*

{{Topological vector spaces

Topological vector spaces