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
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 formula_1 by formula_2 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:
if in addition "X" and "Y" are locally convex then we may add to this list:
if in addition "X" is a seminormed space and "Y" is locally convex then we may add to this list:
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.
For any set "X", the power set of "X" is a bornology on "X" called the discrete 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".
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: 
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. 
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 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".
Limits, products, subspaces, quotients.
Inverse image bornology.
Let "S" be a set, formula_3 be an "I"-indexed family of bounded structures, and let formula_4 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 formula_5. 
Direct image bornology.
Let "S" be a set, formula_3 be an "I"-indexed family of bounded structures, and let formula_4 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 formula_8 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 formula_9 be an "I"-indexed family of bounded structures, let "X" = formula_10, 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 formula_11 