In mathematics, especially functional analysis, a bornology
makes the vector space operations into bounded maps.
that satisfy all the following conditions: Elements of the collection
-bounded or simply bounded sets if
is called a bounded structure or a bornological set.
A base or fundamental system of a bornology
is called the bornology generated by
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
is called a bornological isomorphism.
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 a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then
is called separated if the only bounded vector subspace of
is either the real or complex numbers, in which case a vector bornology
will be called a convex vector bornology if
has a base consisting of convex sets.
of real or complex numbers and
is a topological vector space then the set of all bounded subsets of
called the von Neumann bornology of
and is referred to as natural boundedness.
[1] In any locally convex topological vector space
the set of all closed bounded disks form a base for the usual bornology of
[1] Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.
of real or complex numbers and
forms a neighborhood basis at the origin for a locally convex topological vector space topology.
be the real or complex numbers (endowed with their usual bornologies), let
denote the vector space of all locally bounded
is called the topology of uniform convergence on bounded set.
be the real or complex numbers, and let
denote the vector space of all continuous