Vector bornology

Prerequisites A on a set X is a collection \mathcal{B} of subsets of X that satisfy all the following conditions: • \mathcal{B} covers X; that is, X = \cup \mathcal{B} • \mathcal{B} is stable under inclusions; that is, if B \in \mathcal{B} and A \subseteq B, then A \in \mathcal{B} • \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 {{em|\mathcal{B}-bounded}} 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: • \mathcal{B} is a vector bornology • Finite sums and balanced hulls of \mathcal{B}-bounded sets are \mathcal{B}-bounded • 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: • \mathcal{B} is a vector bornology • addition and scalar multiplication are bounded maps • 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. == 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 . 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 ==