Bornological space

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

• ℬ covers X, i.e. ${\displaystyle X=\bigcup {\mathcal {B}};}$
• ℬ is stable under inclusions, i.e. if A ∈ ℬ and A′ ⊆ A, then A′ ∈ ℬ;
• ℬ is stable under finite unions, i.e. if B₁, ..., B_n ∈ ℬ, then ${\displaystyle \bigcup _{i=1}^{n}B_{i}}$ ∈ ℬ.

Elements of the collection ℬ are usually called ℬ-bounded or simply bounded sets. The pair (X, ℬ) is called a bounded structure' or a bornological set.

A base of the bornology ℬ is a subset ℬ₀ of ℬ such that each element of ℬ is a subset of an element of ℬ₀.

If X is a vector space over a field 𝒦 then a vector bornology on X is a bornology ℬ on X that is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).

If X is a topological vector space (TVS) and ℬ is a bornology on X, then the following are equivalent:

1. ℬ is a vector bornology;
2. finite sums and balanced hulls of ℬ-bounded sets are ℬ-bounded;[1]
3. the scalar multiplication map 𝒦 × X → X defined by (s, x) ↦ sx and the addition map X × X → X defined by (x, y) ↦ x + y, are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).[1]

If in addition ℬ is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then ℬ is called a convex vector bornology. And if the only bounded subspace of X is the trivial subspace (i.e. the space consisting only of ${\displaystyle 0}$) then it is called separated.

A subset A of X is called bornivorous if it absorbs every bounded set. In a vector bornology, A is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology A is bornivorous if it absorbs every bounded disk.

In functional analysis, a bornological space is a locally convex topological vector space whose topology can be recovered from its bornology in a natural way. Explicitly, a Hausdorff locally convex space X with topology ${\displaystyle \tau }$ and continuous dual ${\displaystyle X'}$ is called a bornological space if any one of the following equivalent conditions holds:

• The locally convex topology induced by the von-Neumann bornology on X is the same as ${\displaystyle \tau }$, X's given topology.
• Every convex, balanced, and bornivorous set in X is a neighborhood of zero.
• Every bounded semi-norm on X is continuous,
• Any other Hausdorff locally convex topological vector space topology on X that has the same (von-Neumann) bornology as ${\displaystyle (X,\tau )}$ is necessarily coarser than ${\displaystyle \tau }$.
• For all locally convex spaces Y, every bounded linear operator from X into Y is continuous.
• X is the inductive limit of normed spaces.
• X is the inductive limit of the normed spaces X_D as D varies over the closed and bounded disks of X (or as D varies over the bounded disks of X).
• X carries the Mackey topology ${\displaystyle \tau (X,X')}$ and all bounded linear functionals on X are continuous.
• X has both of the following properties:
  • X is convex-sequential or C-sequential, which means that every convex sequentially open subset of X is open,
  • X is sequentially bornological or S-bornological, which means that every convex and bornivorous subset of X is sequentially open.

where a subset A of X is called sequentially open if every sequence converging to 0 eventually belongs to A.

A disk in a topological vector space X is called infrabornivorous if it absorbs all Banach disks. If X is locally convex and Hausdorff, then a disk is infrabornivorous if and only if it absorbs all compact disks. A locally convex space is called ultrabornological if any of the following conditions hold:

• every infrabornivorous disk is a neighborhood of 0,
• X be the inductive limit of the spaces X_D as D varies over all compact disks in X,
• A seminorm on X that is bounded on each Banach disk is necessarily continuous,
• For every locally convex space Y and every linear map u : X → Y, if u is bounded on each Banach disk then u is continuous.
• For every Banach space Y and every linear map u : X → Y, if u is bounded on each Banach disk then u is continuous.

• Bornology
• Bornivorous set
• Space of linear maps
• Vector bornology

• Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
• Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.