Bornology

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, ${\displaystyle f^{-1}(D)}$ is also bornivorous.[1]

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.[1]

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.[1]

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]

• 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 ${\displaystyle \mathbb {R} }$ form a bornology on ${\displaystyle \mathbb {R} }$. A base for this bornology is given by all closed intervals of the form [-n, n] for n a positive integer.

Let S be a set, ${\displaystyle \left(T_{i},{\mathcal {B}}_{i}\right)_{i\in I}}$ be an I-indexed family of bounded structures, and let ${\displaystyle \left(f_{i},{\mathcal {B}}_{i}\right)_{i\in I}}$ 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 ${\displaystyle \cap _{i\in I}\left[f^{-1}\left({\mathcal {B}}_{i}\right)\right]}$.[1]

Let S be a set, ${\displaystyle \left(T_{i},{\mathcal {B}}_{i}\right)_{i\in I}}$ be an I-indexed family of bounded structures, and let ${\displaystyle \left(f_{i},{\mathcal {B}}_{i}\right)_{i\in I}}$ 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 ${\displaystyle \cup _{i\in I}A_{i}}$ where each A_i ∈ 𝒜_i and all but finitely many A_i are empty.[1]

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]

Let ${\displaystyle \left(X_{i},{\mathcal {B}}_{i}\right)_{i\in I}}$ be an I-indexed family of bounded structures, let X = ${\displaystyle \prod _{i\in I}X_{i}}$, 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 ${\displaystyle \left\{\prod _{i\in I}B_{i}:B_{i}\in {\mathcal {B}}_{i}{\text{ for all }}i\in I\right\}.}$[1]