In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after , the French word for "bounded".

Bornologies and bounded maps

A that satisfy all the following conditions: ; ; ; are called or simply is understood. is called a or a .

A or is called the

are bornological sets then their are both bounded.

Bounded maps

is said to be a or a is called a .

Vector bornologies

be a vector space over a field is called a if it 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.).

is a topological vector space then the following are equivalent: is a vector bornology; -bounded; are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).

is called a if it is stable under the formation of convex hulls is called

will be called a has a base consisting of convex sets.

Bornivorous subsets

is called and a if it absorbs every bounded set.

is bornivorous if it absorbs every bounded disk.

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.

Mackey convergence

is said to be

Bornology of a topological vector space

Every topological vector space at least on a non discrete valued field to be bounded is a locally convex topological vector space

The set of all bounded is called or

is a locally convex topological vector space, then an absorbing disk is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).

Induced topology

of all convex balanced that are bornivorous forms a neighborhood basis at the origin for a locally convex called the .

is a TVS then the

Quasi-bornological spaces

Quasi-bornological spaces where introduced by S. Iyahen in 1968.

A topological vector space with a continuous dual is called a if any of the following equivalent conditions holds:

Every bounded linear operator into another TVS is continuous. into a complete metrizable TVS is continuous. Every knot in a bornivorous string is a neighborhood of the origin.

Every pseudometrizable TVS is quasi-bornological. in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space. into a locally convex bornological space.

Bornological space

In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.

Every locally convex quasi-bornological space is bornological but there exist bornological spaces that are quasi-bornological.

A topological vector space with a continuous dual is called a if it is locally convex and any of the following equivalent conditions holds:

is a neighborhood of zero. Every bounded linear operator into a locally convex TVS is continuous.

• in the domain to a bounded subset of the codomain. In particular, any linear map that is sequentially continuous at the origin is bounded.

into a seminormed space is continuous. into a Banach space is continuous.

is a Hausdorff locally convex space then we may add to this list: The locally convex topology induced by's given topology. Every bounded seminorm is continuous. that has the same (von is the inductive limit of normed spaces. ). are continuous. has both of the following properties:

• is or is open,

• is or is sequentially open.

is called

Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous, where recall that a linear operator is sequentially continuous if and only if it is sequentially continuous at the origin. Thus for linear maps from a bornological space into a locally convex space, continuity is equivalent to sequential continuity at the origin. More generally, we even have the following:

to bounded subsets is necessarily continuous.

Sufficient conditions

As a consequent of the Mackey–Ulam theorem, "for all practical purposes, the product of bornological spaces is bornological."

The following topological vector spaces are all bornological: Any locally convex pseudometrizable TVS is bornological.

• Thus every normed space and Fréchet space is bornological.

Any strict inductive limit of bornological spaces, in particular any strict LF-space, is bornological.

• This shows that there are bornological spaces that are not metrizable.

A countable product of locally convex bornological spaces is bornological. Quotients of Hausdorff locally convex bornological spaces are bornological. The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological. Fréchet Montel spaces have bornological strong duals. The strong dual of every reflexive Fréchet space is bornological. If the strong dual of a metrizable locally convex space is separable, then it is bornological. is bornological. The finest locally convex topology on a vector space is bornological.

1. Counterexamples

There exists a bornological LB-space whose strong bidual is bornological.

A closed vector subspace of a locally convex bornological space is not necessarily bornological. There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.

Bornological spaces need not be barrelled and barrelled spaces need not be bornological. Because every locally convex ultrabornological space is barrelled, it follows that a bornological space is not necessarily ultrabornological.

Properties

The strong dual space of a locally convex bornological space is complete. Every locally convex bornological space is infrabarrelled. Every Hausdorff sequentially complete bornological TVS is ultrabornological.

• Thus every complete Hausdorff bornological space is ultrabornological.

• In particular, every Fréchet space is ultrabornological.

The finite product of locally convex ultrabornological spaces is ultrabornological. Every Hausdorff bornological space is quasi-barrelled. with continuous dual coincides with the Mackey topology

• In particular, bornological spaces are Mackey spaces.

Every quasi-complete (i.e. all closed and bounded subsets are complete) bornological space is barrelled. There exist, however, bornological spaces that are not barrelled. Every bornological space is the inductive limit of normed spaces (and Banach spaces if the space is also quasi-complete). Then the following are equivalent: is bornological. is quasi-barrelled. is barrelled. is a distinguished space. is bornological, then the following are equivalent: is continuous. is sequentially continuous. is bounded. are locally convex is endowed with the topology of uniform convergence on bounded subsets is complete is a complete TVS.

• In particular, the strong dual of a locally convex bornological space is complete. However, it need not be bornological.

• Subsets

is required to be a disk). Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin. Closed vector subspaces of bornological space need not be bornological.

Ultrabornological spaces

is called if it absorbs all Banach disks.

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 if any of the following equivalent conditions hold: Every infrabornivorous disk is a neighborhood of the origin. A seminorm that is bounded on each Banach disk is necessarily continuous. is continuous. is continuous.

Properties

The finite product of ultrabornological spaces is ultrabornological. Inductive limits of ultrabornological spaces are ultrabornological.

See also

References

Bibliography