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.

Definitions

A bornology or boundedness on a set X is a collection ℬ of subsets of X such that

  1. ℬ covers X (i.e. X = );
  2. ℬ is stable under inclusion (i.e. if B ∈ ℬ then every subset of B belongs to ℬ);
  3. ℬ is stable under finite unions (or equivalently, the union of any two sets in ℬ also belongs to ℬ).

in which case the pair (X, ℬ) is called a bounded structure or a bornological set.[1] Elements of ℬ are called ℬ-bounded sets or simply bounded sets, if ℬ is understood. A subset 𝒜 of a bornology ℬ is called a base or fundamental system of ℬ if for every B ∈ ℬ, there exists an A ∈ 𝒜 such that BA. A subset 𝒮 of a bornology ℬ is called a subbase of ℬ if the collection of all finite unions of sets in 𝒮 forms a base for ℬ.[1]

If 𝒜 and ℬ are bornologies on X then we say that ℬ is finer or stronger than 𝒜 and that 𝒜 is coarser or weaker than ℬ if 𝒜 ⊆ ℬ.[1]

If (X, ℬ) is a bounded structure and X ∉ ℬ, then the set of complements { X \ B : B ∈ ℬ } is a filter (having empty intersection) called the filter at infinity.[1]

Given a collection 𝒮 of subsets of X, the smallest bornology containing 𝒮 is called the bornology generated by 𝒮.[1] If f : SX is a map and ℬ is a bornology on X, then we denote the bornology generated by by and called it the inverse image bornology or the initial bornology induced by f on S.[1]

Morphisms: Bounded maps

Suppose that (X, 𝒜) and (Y, ℬ) are bounded structures. A map f : XY is called locally bounded or just bounded if the image under f of every 𝒜-bounded set is a ℬ-bounded set; that is, if for every A ∈ 𝒜, f(A) ∈ ℬ.[1]

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.[1]

Characterizations

Suppose that X and Y are topological vector spaces (TVSs) and f : XY is a linear map. Then the following are equivalent:

  • f is a (locally) bounded map;
  • For every bornivorous (i.e. bounded in the bornological sense) disk D in Y, is also bornivorous.[1]

if in addition X and Y are locally convex then we may add to this list:

  1. f takes bounded disks to bounded disks;

if in addition X is a seminormed space and Y is locally convex then we may add to this list:

  1. f maps null sequences (i.e. sequences converging to 0) into bounded subsets of Y.[1]

Examples

If X and Y are any two topological vector spaces (they need not even be Hausdorff) and if f : X → Y is a continuous linear operator between them, then f is a bounded linear operator (when X and Y have their von-Neumann bornologies). The converse is in general false.

A sequentially continuous map f : XY between two TVSs is necessarily locally bounded.[1]

Examples and sufficient conditions

Discrete bornology

For any set X, the power set of X is a bornology on X called the discrete bornology.[1]

Compact bornology

For any topological space X, the set of all relatively compact subsets of X form a bornology on X called the compact bornology on X.[1]

Closure and interior bornologies

Suppose that X is a topological space and ℬ is a bornology on X. The bornology generated by the set of all interiors of sets in ℬ (i.e. by { int B : B ∈ ℬ }) is called the interior of ℬ and is denoted by int ℬ.[1] The bornology ℬ is called open if ℬ = int ℬ. Then the bornology generated by the set of all closures of sets in ℬ (i.e. by { cl B : B ∈ ℬ }) is called the closure of ℬ and is denoted by cl ℬ.[1] We necessarily have int ℬ ⊆ ℬ ⊆ cl ℬ.

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

  1. ℬ = cl ℬ;
  2. the closed subsets of X generate ℬ;[1]
  3. the closure of every B ∈ ℬ belongs to ℬ.[1]

The bornology ℬ is called proper if ℬ is both open and closed.[1]

The topological space X is called locally ℬ-bounded or just locally bounded if every xX has a neighborhood that belongs to ℬ. Every compact subset of a locally bounded topological space is bounded.[1]

Topological rings

Suppose that X is a commutative topological ring. A subset S of X is called bounded if for each neighborhood U of 0 in X, there exists a neighborhood V of 0 in X such that SVU.[1]

Bornology on a topological vector space

If X is a topological vector space (TVS) then the set of all boundedness subsets of X form a bornology (indeed, even a vector bornology) on X called the von Neumann bornology of X, the usual bornology, or simply the bornology of X and is referred to as natural boundedness.[1] In any locally convex TVS X, the set of all closed bounded disks form a base for the usual bornology of X.[1]

Trivial examples

  • For any set X, the set of all finite subsets of X is a bornology on X; the same is true of the set of all countable subsets of X.
    • More generally, for any infinite cardinal 𝜅, the set of all subsets of X having cardinality at most 𝜅 is a bornology on X.
  • 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 [-n, n] for n a positive integer.

Limits, products, subspaces, quotients

Inverse image bornology

Let S be a set, be an I-indexed family of bounded structures, and let be an I-indexed family of maps where for all iI, fi : STi. The inverse image bornology 𝒜 on S determined by these maps is the strongest bornology on S making each fi : (S, 𝒜) → (Ti, ℬi) locally bounded. This bornology is equal to .[1]

Direct image bornology

Let S be a set, be an I-indexed family of bounded structures, and let be an I-indexed family of maps where for all iI, fi : TiS. The direct image bornology 𝒜 on S determined by these maps is the weakest bornology on S making each fi : (Ti, ℬi) → (S, 𝒜) locally bounded. If for each iI, 𝒜i denotes the bornology generated by f(ℬi), then this bornology is equal to the collection of all subsets A of S of the form where each Ai ∈ 𝒜i and all but finitely many Ai are empty.[1]

Subspace bornology

Suppose that (X, ℬ) is a bounded structure and S be a subset of X. The subspace bornology 𝒜 on S is the finest bornology on S making the natural inclusion map of S into X, Id : (S, 𝒜) → (X, ℬ), locally bounded.[1]

Product bornology

Let be an I-indexed family of bounded structures, let X = , and for each iI, let fi : XXi denote the canonical projection. The product bornology on X is the inverse image bornology determined by the canonical projections fi : XXi. That is, it is the strongest bornology on X making each of the canonical projections locally bounded. A base for the product bornology is given by [1]

See also

References

  1. Narici 2011, pp. 156-175.
  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 978-1-58488-866-6. OCLC 144216834.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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