Bornology

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. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is because^{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.

History

Bornology originates from functional analysis. There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies and the other is to study notions related to boundedness.
For normed spaces, from which functional analysis arose, topological and bornological notions are distinct but complementary and closely related.
For example, the unit ball centered at the origin is both a neighborhood of the origin and a bounded subset.
Furthermore, a subset of a normed space is a neighborhood of the origin exactly when it contains a non-zero scalar multiple of this ball; so this is one instance where the topological and bornological notions are distinct but complementary.
Other times, the distinction between topological and bornological notions may even be unnecessary.
For example, for linear maps between normed spaces, being continuous is equivalent to being bounded.
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.
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.
Born from the work of George Mackey, the importance of bounded subsets first became apparent in duality theory, especially because of the Mackey–Arens theorem and the Mackey topology.
Starting around the 1950s, it became apparent that topological vector spaces were inadequate for the study of certain major problems.
For example, the multiplication operation of some important topological algebras was not continuous, although it was often bounded.
Other major problems for which TVSs were found to be inadequate was in developing a more general theory of differential calculus, generalizing distributions from scalar-valued distributions to vector or operator-valued distributions, and extending the holomorphic functional calculus of Gelfand to a broader class of operators, including those whose spectra are not compact.
Bornology has been found to be a useful tool for investigating these problems and others, including problems in algebraic geometry and general topology.

Definitions

A on a set is a cover of the set that is closed under finite unions and taking subsets. Elements of a bornology are called.
Explicitly, a or on a set is a family of subsets of such that

is stable under inclusion or : If then every subset of is an element of
- Stated in plain English, this says that subsets of bounded sets are bounded.
covers Every point of is an element of some or equivalently,
- Assuming, this condition may be replaced with: For every In plain English, this says that every point is bounded.
is stable under finite unions: The union of finitely many elements of is an element of or equivalently, the union of any sets belonging to also belongs to
- In plain English, this says that the union of two bounded sets is a bounded set.

in which case the pair is called a or a.
Thus a bornology can equivalently be defined as a [|downward closed] cover that is closed under binary unions.
A non-empty family of sets that closed under finite unions and taking subsets and ) is called an . A bornology on a set can thus be equivalently defined as an ideal that covers
Elements of are called or simply, if is understood.
Properties and imply that every singleton subset of is an element of every bornology on property, 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. In particular, the empty set is always bounded.
If is a bounded structure and then the set of complements is a filter called the ; it is always a, which by definition means that it has empty intersection/kernel, because for every

Bases and subbases

If and are bornologies on then is said to be or than and also is said to be or than if
A family of sets is called a or of a bornology if and for every there exists an such that
A family of sets is called a of a bornology if and the collection of all finite unions of sets in forms a base for
Every base for a bornology is also a subbase for it.

Generated bornology

The intersection of any collection of bornologies on is once again a bornology on
Such an intersection of bornologies will cover because every bornology on contains every finite subset of . It is readily verified that such an intersection will also be closed under inclusion and finite unions and thus will be a bornology on
Given a collection of subsets of the smallest bornology on containing is called the.
It is equal to the intersection of all bornologies on that contain as a subset.
This intersection is well-defined because the power set of is always a bornology on so every family of subsets of is always contained in at least one bornology on

Bounded maps

Suppose that and are bounded structures.
A map is called a, or just a, if the image under of every -bounded set is a -bounded set;
that is, if for every
Since the composition of two locally bounded map is again locally bounded, it is clear that the class of all bounded structures forms a category whose morphisms are bounded maps.
An isomorphism in this category is called a and it is a bijective locally bounded map whose inverse is also locally bounded.

Examples of bounded maps

If is a continuous linear operator between two topological vector spaces, then it is a bounded linear operator when and have their von-Neumann bornologies, where a set is bounded precisely when it is absorbed by all neighbourhoods of origin.
The converse is in general false.
A sequentially continuous map between two TVSs is necessarily locally bounded.

General constructions

Discrete bornology
For any set the power set of is a bornology on called the. Since every bornology on is a subset of the discrete bornology is the finest bornology on
If is a bounded structure then is the discrete bornology if and only if
Indiscrete bornology
For any set the set of all finite subsets of is a bornology on called the. It is the coarsest bornology on meaning that it is a subset of every bornology on
Sets of bounded cardinality
The set of all countable subsets of is a bornology on
More generally, for any infinite cardinal the set of all subsets of having cardinality at most is a bornology on

Inverse image bornology

If is a map and is a bornology on then denotes the bornology generated by which is called it the or the induced by on
Let be a set, be an -indexed family of bounded structures, and let be an -indexed family of maps where for every
The on determined by these maps is the strongest bornology on making each locally bounded.
This bornology is equal to

Direct image bornology

Let be a set, be an -indexed family of bounded structures, and let be an -indexed family of maps where for every
The on determined by these maps is the weakest bornology on making each locally bounded.
If for each denotes the bornology generated by then this bornology is equal to the collection of all subsets of of the form where each and all but finitely many are empty.

Subspace bornology

Suppose that is a bounded structure and be a subset of
The on is the finest bornology on making the inclusion map of into locally bounded.

Product bornology

Let be an -indexed family of bounded structures, let and for each let denote the canonical projection.
The on is the inverse image bornology determined by the canonical projections
That is, it is the strongest bornology on making each of the canonical projections locally bounded.
A base for the product bornology is given by

Topological constructions

Compact bornology

A subset of a topological space is called relatively compact if its closure is a compact subspace of
For any topological space in which singleton subsets are relatively compact, the set of all relatively compact subsets of form a bornology on called the on
Every continuous map between T1 spaces is bounded with respect to their compact bornologies.
The set of relatively compact subsets of form a bornology on A base for this bornology is given by all closed intervals of the form for

Metric bornology

Given a metric space the consists of all subsets such that the supremum is finite.
Similarly, given a measure space the family of all measurable subsets of finite measure form a bornology on

Closure and interior bornologies

Suppose that is a topological space and is a bornology on
The bornology generated by the set of all topological interiors of sets in is called the of and is denoted by
We necessarily have
The bornology is called if it satisfies any of the following equivalent conditions:

the closed subsets of generate ;
the closure of every belongs to

The bornology is called if is both open and closed.
The topological space is called or just if every has a neighborhood that belongs to
Every compact subset of a locally bounded topological space is bounded.