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.
[citation needed] The name was coined by Bourbaki[citation needed] after borné, the French word for "bounded".
is called a bounded structure or a bornological set.
is the bornology having as a base the collection of all sets of the form
is bounded in the product bornology if and only if its image under the canonical projections onto
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 called separated if the only bounded vector subspace of
is either the real or complex numbers, in which case a vector bornology
is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology
Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.
[3] Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.
if there exists a sequence of positive real numbers
is a locally convex topological vector space then
The set of all bounded subsets of a topological vector space
is a locally convex topological vector space, then an absorbing disk
infrabornivorous) if and only if its Minkowski functional is locally bounded (resp.
that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on
endowed with the locally convex topology induced by the von Neumann bornology of
endowed with the topology induced by von Neumann bornology of
in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.
is a quasi-bornological TVS then the finest locally convex topology on
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 not quasi-bornological.
is called a bornological space if it is locally convex and any of the following equivalent conditions holds: If
is a Hausdorff locally convex space then we may add to this list:[7] Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous,[4] 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.
[4][14] There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.
is called infrabornivorous if it absorbs all Banach disks.