In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness.
This is because[1]pg 9 the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra.
There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies (vector topologies, continuous operators, open/compact subsets, etc.)
For normed spaces, from which functional analysis arose, topological and bornological notions are distinct but complementary and closely related.
Furthermore, a subset of a normed space is a neighborhood of the origin (respectively, is a bounded set) exactly when it contains (respectively, it is contained in) a non-zero scalar multiple of this ball; so this is one instance where the topological and bornological notions are distinct but complementary (in the sense that their definitions differ only by which of
Other times, the distinction between topological and bornological notions may even be unnecessary.
Although the distinction between topology and bornology is often blurred or unnecessary for normed space, it becomes more important when studying generalizations of normed spaces.
Nevertheless, bornology and topology can still be thought of as two necessary, distinct, and complementary aspects of one and the same reality.
[2] The general theory of topological vector spaces arose first from the theory of normed spaces and then bornology emerged from this general theory of topological vector spaces, although bornology has since become recognized as a fundamental notion in functional analysis.
[3] Starting around the 1950s, it became apparent that topological vector spaces were inadequate for the study of certain major problems.
[3] For example, the multiplication operation of some important topological algebras was not continuous, although it was often bounded.
[3] Other major problems for which TVSs were found to be inadequate was in developing a more general theory of differential calculus, generalizing distributions from (the usual) scalar-valued distributions to vector or operator-valued distributions, and extending the holomorphic functional calculus of Gelfand (which is primarily concerted with Banach algebras or locally convex algebras) to a broader class of operators, including those whose spectra are not compact.
Elements of a bornology are called bounded sets.
is called a bounded structure or a bornological set.
A non-empty family of sets that closed under finite unions and taking subsets (properties (1) and (3)) is called an ideal (because it is an ideal in the Boolean algebra/field of sets consisting of all subsets).
property (3), in turn, guarantees that the same is true of every finite subset of
In other words, points and finite subsets are always bounded in every bornology.
is called a base or fundamental system of a bornology
It is readily verified that such an intersection will also be closed under (subset) inclusion and finite unions and thus will be a bornology on
An isomorphism in this category is called a bornomorphism and it is a bijective locally bounded map whose inverse is also locally bounded.
is a continuous linear operator between two topological vector spaces (not necessarily Hausdorff), then it is a bounded linear operator when
have their von-Neumann bornologies, where a set is bounded precisely when it is absorbed by all neighbourhoods of origin (these are the subsets of a TVS that are normally called bounded when no other bornology is explicitly mentioned.).
is a bounded structure then (because bornologies are downward closed)
is the inverse image bornology determined by the canonical projections
making each of the canonical projections locally bounded.
[5] Every continuous map between T1 spaces is bounded with respect to their compact bornologies.
A base for this bornology is given by all closed intervals of the form
is called closed if it satisfies any of the following equivalent conditions: The bornology
is a topological vector space (TVS) then the set of all bounded subsets of
the set of all closed bounded disks forms a base for the usual bornology of