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[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.

History

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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 (vector topologies, continuous operators, open/compact subsets, etc.) and the other is to study notions related to boundedness[2] (vector bornologies, bounded operators, bounded subsets, etc.).

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 (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   and   is used). Other times, the distinction between topological and bornological notions may even be unnecessary. For example, for linear maps between normed spaces, being continuous (a topological notion) is equivalent to being bounded (a bornological notion). 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] Born from the work of George Mackey (after whom Mackey spaces are named), the importance of bounded subsets first became apparent in duality theory, especially because of the Mackey–Arens theorem and the Mackey topology.[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. Bornology has been found to be a useful tool for investigating these problems and others,[4] including problems in algebraic geometry and general topology.

Definitions

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A bornology on a set is a cover of the set that is closed under finite unions and taking subsets. Elements of a bornology are called bounded sets.

Explicitly, a bornology or boundedness on a set   is a family   of subsets of   such that

  1.   is stable under inclusion or downward closed: If   then every subset of   is an element of  
    • Stated in plain English, this says that subsets of bounded sets are bounded.
  2.   covers   Every point of   is an element of some   or equivalently,  
    • Assuming (1), this condition may be replaced with: For every     In plain English, this says that every point is bounded.
  3.   is stable under finite unions: The union of finitely many elements of   is an element of   or equivalently, the union of any two 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 bounded structure or a bornological set.[5]

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 (properties (1) and (3)) is called an ideal (because it is an ideal in the Boolean algebra/field of sets consisting of all subsets). A bornology on a set   can thus be equivalently defined as an ideal that covers  

Elements of   are called  -bounded sets or simply bounded sets, if   is understood. Properties (1) and (2) imply that every singleton subset of   is an element of every bornology on   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. In particular, the empty set is always bounded.

If   is a bounded structure and   then the set of complements   is a (proper) filter called the filter at infinity;[5] it is always a free filter, which by definition means that it has empty intersection/kernel, because   for every  

Bases and subbases

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If   and   are bornologies on   then   is said to be finer or stronger than   and also   is said to be coarser or weaker than   if  [5]

A family of sets   is called a base or fundamental system of a bornology   if   and for every   there exists an   such that  

A family of sets   is called a subbase of a bornology   if   and the collection of all finite unions of sets in   forms a base for  [5]

Every base for a bornology is also a subbase for it.

Generated bornology

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The intersection of any collection of (one or more) bornologies on   is once again a bornology on   Such an intersection of bornologies will cover   because every bornology on   contains every finite subset of   (that is, if   is a bornology on   and   is finite then  ). 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  

Given a collection   of subsets of   the smallest bornology on   containing   is called the bornology generated by  .[5] 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

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Suppose that   and   are bounded structures. A map   is called a locally bounded map, or just a bounded map, if the image under   of every  -bounded set is a  -bounded set; that is, if for every    [5]

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 bornomorphism and it is a bijective locally bounded map whose inverse is also locally bounded.[5]

Examples of bounded maps

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If   is a continuous linear operator between two topological vector spaces (not necessarily Hausdorff), 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 (these are the subsets of a TVS that are normally called bounded when no other bornology is explicitly mentioned.). The converse is in general false.

A sequentially continuous map   between two TVSs is necessarily locally bounded.[5]

General constructions

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Discrete bornology

For any set   the power set   of   is a bornology on   called the discrete bornology.[5] Since every bornology on   is a subset of   the discrete bornology is the finest bornology on   If   is a bounded structure then (because bornologies are downward closed)   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 indiscrete bornology. 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

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If   is a map and   is a bornology on   then   denotes the bornology generated by   which is called it the inverse image bornology or the initial bornology induced by   on  [5]

Let   be a set,   be an  -indexed family of bounded structures, and let   be an  -indexed family of maps where   for every   The inverse image bornology   on   determined by these maps is the strongest bornology on   making each   locally bounded. This bornology is equal to[5]  

Direct image bornology

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Let   be a set,   be an  -indexed family of bounded structures, and let   be an  -indexed family of maps where   for every   The direct image bornology   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.[5]

Subspace bornology

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Suppose that   is a bounded structure and   be a subset of   The subspace bornology   on   is the finest bornology on   making the inclusion map   of   into   (defined by  ) locally bounded.[5]

Product bornology

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Let   be an  -indexed family of bounded structures, let   and for each   let   denote the canonical projection. The product bornology 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  [5]

Topological constructions

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Compact bornology

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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 (such as a T1 space), the set of all relatively compact subsets of   form a bornology on   called the compact bornology on  [5] 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

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Given a metric space   the metric bornology consists of all subsets   such that the supremum   is finite.

Similarly, given a measure space   the family of all measurable subsets   of finite measure (meaning  ) form a bornology on  

Closure and interior bornologies

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Suppose that   is a topological space and   is a bornology on  

The bornology generated by the set of all topological interiors of sets in   (that is, generated by   is called the interior of   and is denoted by  [5] The bornology   is called open if  

The bornology generated by the set of all topological closures of sets in   (that is, generated by  ) is called the closure of   and is denoted by  [5] We necessarily have  

The bornology   is called closed if it satisfies any of the following equivalent conditions:

  1.  
  2. the closed subsets of   generate  ;[5]
  3. the closure of every   belongs to  [5]

The bornology   is called proper if   is both open and closed.[5]

The topological space   is called locally  -bounded or just locally bounded if every   has a neighborhood that belongs to   Every compact subset of a locally bounded topological space is bounded.[5]

Bornology of a topological vector space

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If   is a topological vector space (TVS) then the set of all bounded subsets of   form a bornology (indeed, even a vector bornology) on   called the von Neumann bornology of  , the usual bornology, or simply the bornology of   and is referred to as natural boundedness.[5] In any locally convex TVS   the set of all closed bounded disks forms a base for the usual bornology of  [5]

A linear map between two bornological spaces is continuous if and only if it is bounded (with respect to the usual bornologies).

Topological rings

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Suppose that   is a commutative topological ring. A subset   of   is called a bounded set if for each neighborhood   of the origin in   there exists a neighborhood   of the origin in   such that  [5]

See also

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References

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  1. ^ Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection". arXiv:0803.1529 [math.QA].
  2. ^ a b Hogbe-Nlend 1971, p. 5.
  3. ^ a b c d Hogbe-Nlend 1971, pp. 1–2.
  4. ^ Hogbe-Nlend 1971.
  5. ^ a b c d e f g h i j k l m n o p q r s t u v w x Narici & Beckenstein 2011, pp. 156–175.