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.

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 = $\cup _{B\in {\mathcal {B}}}B$);
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. 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 B ⊆ A. 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.

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.

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.

Given a collection 𝒮 of subsets of X, the smallest bornology containing 𝒮 is called the bornology generated by 𝒮. If f : S → X is a map and ℬ is a bornology on X, then we denote the bornology generated by $f^{-1}\left({\mathcal {B}}\right):=\left\{f^{-1}\left(B\right):B\in {\mathcal {B}}\right\}$ by $\left[f^{-1}\left({\mathcal {B}}\right)\right]$ and called it the inverse image bornology or the initial bornology induced by f on S.

Morphisms: Bounded maps

Suppose that (X, 𝒜) and (Y, ℬ) are bounded structures. A map f : X → Y 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.

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.

Characterizations

Suppose that X and Y are topological vector spaces (TVSs) and f : X → Y 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, $f^{-1}(D)$ is also bornivorous.

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

3. 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:

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

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 : X → Y between two TVSs is necessarily locally bounded.

Examples and sufficient conditions

Discrete bornology

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

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.

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. 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. 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;
3. the closure of every B ∈ ℬ belongs to ℬ.

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

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

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 SV ⊆ U.

Bornology on a topological vector space

If X is a(n indiscrete) topological vector space (TVS) then the set of all bounded 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. In any locally convex TVS X, the set of all closed bounded disks form a base for the usual bornology of X.

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 $\mathbb {R}$ form a bornology on $\mathbb {R}$. 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, $\left(T_{i},{\mathcal {B}}_{i}\right)_{i\in I}$ be an I-indexed family of bounded structures, and let $\left(f_{i},{\mathcal {B}}_{i}\right)_{i\in I}$ be an I-indexed family of maps where for all i ∈ I, f_i : S → T_i. The inverse image bornology 𝒜 on S determined by these maps is the strongest bornology on S making each f_i : (S, 𝒜) → (T_i, ℬ_i) locally bounded. This bornology is equal to $\cap _{i\in I}\left[f^{-1}\left({\mathcal {B}}_{i}\right)\right]$.

Direct image bornology

Let S be a set, $\left(T_{i},{\mathcal {B}}_{i}\right)_{i\in I}$ be an I-indexed family of bounded structures, and let $\left(f_{i},{\mathcal {B}}_{i}\right)_{i\in I}$ be an I-indexed family of maps where for all i ∈ I, f_i : T_i → S. The direct image bornology 𝒜 on S determined by these maps is the weakest bornology on S making each f_i : (T_i, ℬ_i) → (S, 𝒜) locally bounded. If for each i ∈ I, 𝒜_i denotes the bornology generated by f(ℬ_i), then this bornology is equal to the collection of all subsets A of S of the form $\cup _{i\in I}A_{i}$ where each A_i ∈ 𝒜_i and all but finitely many A_i are empty.

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.

Product bornology

Let $\left(X_{i},{\mathcal {B}}_{i}\right)_{i\in I}$ be an I-indexed family of bounded structures, let X = $\prod _{i\in I}X_{i}$, and for each i ∈ I, let f_i : X → X_i denote the canonical projection. The product bornology on X is the inverse image bornology determined by the canonical projections f_i : X → X_i. 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 $\left\{\prod _{i\in I}B_{i}:B_{i}\in {\mathcal {B}}_{i}{\text{ for all }}i\in I\right\}.$

See also

• Bornivorous set
• Bornological space
• Space of linear maps
• Ultrabornological space
• Vector bornology

References

• 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. Vol. 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. Vol. 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.