Vector bornology

In mathematics, especially functional analysis, a bornology ${ℬ}$ 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 ${ℬ}$ makes the vector space operations into bounded maps.

Definitions

Prerequisits

Main article: Bornology

A bornology 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 ${ℬ}$-bounded or simply bounded sets if ${ℬ}$ is understood. The pair $(X,{ℬ})$ is called a bounded structure or a bornological set.

A base or fundamental system 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 ${𝒮}.$

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

Vector bornology

Let $X$ be a vector space over a field $\mathbb{𝕂}$ where $\mathbb{𝕂}$ has a bornology ${{ℬ}}_{\mathbb{𝕂}}.$ A bornology ${ℬ}$ on $X$ is called a 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 ${ℬ}$ 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 ${ℬ}$-bounded
3. 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)

A vector bornology ${ℬ}$ 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 ${ℬ}.$ And a vector bornology ${ℬ}$ is called 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 ${ℬ}$ on $X$ will be called a convex vector bornology if ${ℬ}$ 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 ${ℬ}$ is a bornology on $X.$ Then the following are equivalent:

1. ${ℬ}$ is a vector bornology
2. addition and scalar multiplication are bounded maps
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 ${ℬ}$

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 von Neumann bornology of $X$, the usual bornology, or simply the bornology of $X$ and is referred to as natural boundedness. 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 $\mathbb{𝕂}$ 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 $\mathbb{𝕂}$ be the real or complex numbers (endowed with their usual bornologies), let $(T,{ℬ})$ 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 $B\in {ℬ},$ let $p_B(f):=\sup |f(B)|$ 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 $\{p_B:B\in {ℬ}\}$ is called the topology of uniform convergence on bounded set. This topology makes $LB(T,\mathbb{𝕂})$ into a complete space.

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 $C(T,\mathbb{𝕂}).$

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.