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

Main article: Bornology

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

1. ${\mathcal {B}}$ covers $X;$ that is, $X=\cup {\mathcal {B}}$
2. ${\mathcal {B}}$ is stable under inclusions; that is, if $B\in {\mathcal {B}}$ and $A\subseteq B,$ then $A\in {\mathcal {B}}$
3. ${\mathcal {B}}$ 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 ${\mathcal {B}}$-bounded or simply bounded sets if ${\mathcal {B}}$ is understood. The pair $(X,{\mathcal {B}})$ is called a bounded structure or a bornological set.

A base or 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 ${\mathcal {S}}.$

If $(X,{\mathcal {B}})$ and $(Y,{\mathcal {C}})$ 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 $B\in {\mathcal {B}}$ and $C\in {\mathcal {C}}.$ 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 $(X,{\mathcal {B}})$ and $(Y,{\mathcal {C}})$ 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 ${\mathcal {B}}$-bounded subsets of $X$ to ${\mathcal {C}}$-bounded subsets of $Y;$ that is, if $f\left({\mathcal {B}}\right)\subseteq {\mathcal {C}}.$ 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 ${\mathcal {B}}_{\mathbb {K} }.$ A bornology ${\mathcal {B}}$ 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 ${\mathcal {B}}$ is a bornology on $X,$ then the following are equivalent:

1. ${\mathcal {B}}$ is a vector bornology
2. Finite sums and balanced hulls of ${\mathcal {B}}$-bounded sets are ${\mathcal {B}}$-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 ${\mathcal {B}}$ 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 ${\mathcal {B}}.$ And a vector bornology ${\mathcal {B}}$ 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 ${\mathcal {B}}$ on $X$ will be called a 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:

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

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 ${\mathcal {B}}$ 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 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

• Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
• 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.