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.

Prerequisits
A 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 or simply  if $$\mathcal{B}$$ is understood. The pair $$(X, \mathcal{B})$$ is called a or a.

A or  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 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 or a  (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.

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 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 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 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 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, 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 $$\mathbb{K}$$ 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.

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. 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 $$C(T, \mathbb{K}).$$