Vector bornology explained
on a
vector space
over a
field
where
has a bornology ℬ
, is called a
vector bornology if
makes the vector space operations into bounded maps.
Definitions
Prerequisits
See main article: Bornology.
A on a set
is a collection
of subsets of
that satisfy all the following conditions:
covers
that is,
is stable under inclusions; that is, if
and
then
is stable under finite unions; that is, if
then
Elements of the collection
are called or simply if
is understood. The pair
is called a or a .
A or of a bornology
is a subset
of
such that each element of
is a subset of some element of
Given a collection
of subsets of
the smallest bornology containing
is called the
bornology generated by
If
and
are bornological sets then their on
is the bornology having as a base the collection of all sets of the form
where
and
A subset of
is bounded in the product bornology if and only if its image under the canonical projections onto
and
are both bounded.
If
and
are bornological sets then a function
is said to be a or a (with respect to these bornologies) if it maps
-bounded subsets of
to
-bounded subsets of
that is, if
f\left(l{B}\right)\subseteql{C}.
If in addition
is a bijection and
is also bounded then
is called a .
Vector bornology
Let
be a vector space over a
field
where
has a bornology
A bornology
on
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.).
If
is a vector space and
is a bornology on
then the following are equivalent:
is a vector bornology
- Finite sums and balanced hulls of
-bounded sets are
-bounded
- The scalar multiplication map
defined by
and the addition map
defined by
are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets)
A vector bornology
is called a if it is stable under the formation of
convex hulls (i.e. the convex hull of a bounded set is bounded) then
And a vector bornology
is called if the only bounded vector subspace of
is the 0-dimensional trivial space
Usually,
is either the real or complex numbers, in which case a vector bornology
on
will be called a if
has a base consisting of
convex sets.
Characterizations
Suppose that
is a
vector space over the field
of real or complex numbers and
is a bornology on
Then the following are equivalent:
is a vector bornology
- addition and scalar multiplication are bounded maps
- the balanced hull of every element of
is an element of
and the sum of any two elements of
is again an element of
Bornology on a topological vector space
If
is a topological vector space then the set of all
bounded subsets of
from a vector bornology on
called the, the, or simply the of
and is referred to as .In any
locally convex topological vector space
the set of all closed bounded
disks form a base for the usual bornology of
Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.
Topology induced by a vector bornology
Suppose that
is a vector space over the field
of real or complex numbers and
is a vector bornology on
Let
denote all those subsets
of
that are convex,
balanced, and
bornivorous.Then
forms a neighborhood basis at the origin for a
locally convex topological vector space topology.
Examples
Locally convex space of bounded functions
Let
be the real or complex numbers (endowed with their usual bornologies), let
be a bounded structure, and let
denote the vector space of all locally bounded
-valued maps on
For every
let
pB(f):=\sup\left|f(B)\right|
for all
where this defines a
seminorm on
The
locally convex topological vector space topology on
defined by the family of seminorms
\left\{pB:B\inl{B}\right\}
is called the .This topology makes
into a
complete space.
Bornology of equicontinuity
Let
be a topological space,
be the real or complex numbers, and let
denote the vector space of all continuous
-valued maps on
The set of all
equicontinuous subsets of
forms a vector bornology on
See also
Bibliography
- Book: The Convenient Setting of Global Analysis . Kriegl . Andreas . Michor . Peter W. . 1997 . . Mathematical Surveys and Monographs . 978-082180780-4.
