Vector bornology

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

1. ${\displaystyle {\mathcal {B}}}$ covers ${\displaystyle X;}$ that is, ${\displaystyle X=\cup {\mathcal {B}}}$
2. ${\displaystyle {\mathcal {B}}}$ is stable under inclusions; that is, if ${\displaystyle B\in {\mathcal {B}}}$ and ${\displaystyle A\subseteq B,}$ then ${\displaystyle A\in {\mathcal {B}}}$
3. ${\displaystyle {\mathcal {B}}}$ is stable under finite unions; that is, if ${\displaystyle B_{1},\ldots ,B_{n}\in {\mathcal {B}}}$ then ${\displaystyle B_{1}\cup \cdots \cup B_{n}\in {\mathcal {B}}}$

Elements of the collection ${\displaystyle {\mathcal {B}}}$ are called ${\displaystyle {\mathcal {B}}}$-bounded or simply bounded sets if ${\displaystyle {\mathcal {B}}}$ is understood. The pair ${\displaystyle (X,{\mathcal {B}})}$ is called a bounded structure or a bornological set.

A base or fundamental system of a bornology ${\displaystyle {\mathcal {B}}}$ is a subset ${\displaystyle {\mathcal {B}}_{0}}$ of ${\displaystyle {\mathcal {B}}}$ such that each element of ${\displaystyle {\mathcal {B}}}$ is a subset of some element of ${\displaystyle {\mathcal {B}}_{0}.}$ Given a collection ${\displaystyle {\mathcal {S}}}$ of subsets of ${\displaystyle X,}$ the smallest bornology containing ${\displaystyle {\mathcal {S}}}$ is called the bornology generated by ${\displaystyle {\mathcal {S}}.}$[1]

If ${\displaystyle (X,{\mathcal {B}})}$ and ${\displaystyle (Y,{\mathcal {C}})}$ are bornological sets then their product bornology on ${\displaystyle X\times Y}$ is the bornology having as a base the collection of all sets of the form ${\displaystyle B\times C,}$ where ${\displaystyle B\in {\mathcal {B}}}$ and ${\displaystyle C\in {\mathcal {C}}.}$[1] A subset of ${\displaystyle X\times Y}$ is bounded in the product bornology if and only if its image under the canonical projections onto ${\displaystyle X}$ and ${\displaystyle Y}$ are both bounded.

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

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

1. ${\displaystyle {\mathcal {B}}}$ is a vector bornology
2. Finite sums and balanced hulls of ${\displaystyle {\mathcal {B}}}$-bounded sets are ${\displaystyle {\mathcal {B}}}$-bounded[1]
3. The scalar multiplication map ${\displaystyle \mathbb {K} \times X\to X}$ defined by ${\displaystyle (s,x)\mapsto sx}$ and the addition map ${\displaystyle X\times X\to X}$ defined by ${\displaystyle (x,y)\mapsto x+y,}$ are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets)[1]

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

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

Let ${\displaystyle \mathbb {K} }$ be the real or complex numbers (endowed with their usual bornologies), let ${\displaystyle (T,{\mathcal {B}})}$ be a bounded structure, and let ${\displaystyle LB(T,\mathbb {K} )}$ denote the vector space of all locally bounded ${\displaystyle \mathbb {K} }$-valued maps on ${\displaystyle T.}$ For every ${\displaystyle B\in {\mathcal {B}},}$ let ${\displaystyle p_{B}(f):=\sup \left|f(B)\right|}$ for all ${\displaystyle f\in LB(T,\mathbb {K} ),}$ where this defines a seminorm on ${\displaystyle X.}$ The locally convex topological vector space topology on ${\displaystyle LB(T,\mathbb {K} )}$ defined by the family of seminorms ${\displaystyle \left\{p_{B}:B\in {\mathcal {B}}\right\}}$ is called the topology of uniform convergence on bounded set.[1] This topology makes ${\displaystyle LB(T,\mathbb {K} )}$ into a complete space.[1]

Let ${\displaystyle T}$ be a topological space, ${\displaystyle \mathbb {K} }$ be the real or complex numbers, and let ${\displaystyle C(T,\mathbb {K} )}$ denote the vector space of all continuous ${\displaystyle \mathbb {K} }$-valued maps on ${\displaystyle T.}$ The set of all equicontinuous subsets of ${\displaystyle C(T,\mathbb {K} )}$ forms a vector bornology on ${\displaystyle C(T,\mathbb {K} ).}$[1]