In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, 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 mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, in the same way that a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
spaces is continuous if and only if it is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
.
Bornological spaces were first studied by
George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Un ...
. The name was coined by
Bourbaki after , the French word for "
bounded".
Bornologies and bounded maps
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
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.
Bounded maps
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
If in addition
is a bijection and
is also bounded then
is called a .
Vector bornologies
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 hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \l ...
s (i.e. if the sum of two bounded sets is bounded, etc.).
If
is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) 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 hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
s (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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
sets.
Bornivorous subsets
A subset
of
is called and a if it
absorbs every bounded set.
In a vector bornology,
is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology
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
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
is a neighborhood of the origin.
Mackey convergence
A sequence
in a TVS
is said to be if there exists a sequence of positive real numbers
diverging to
such that
converges to
in
Bornology of a topological vector space
Every
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
at least on a non discrete
valued field
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
gives a bornology on
by defining a subset
to be
bounded (or von-Neumann bounded), if and only if for all open sets
containing zero there exists a
with
If
is a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
then
is bounded if and only if all continuous semi-norms on
are bounded on
The set of all
bounded subsets of a topological vector space
is called or of
If
is a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
, then an
absorbing disk in
is bornivorous (resp. infrabornivorous) if and only if its
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, t ...
is locally bounded (resp. infrabounded).
Induced topology
If
is a convex vector bornology on a vector space
then the collection
of all convex
balanced subsets of
that are bornivorous forms a
neighborhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
at the origin for a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
topology on
called the .
If
is a TVS then the is the vector space
endowed with the locally convex topology induced by the von Neumann bornology of
Quasi-bornological spaces
Quasi-bornological spaces where introduced by S. Iyahen in 1968.
A
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS)
with a
continuous dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
is called a if any of the following equivalent conditions holds:
- Every
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
from into another TVS is continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
.
- Every bounded linear operator from into a complete metrizable TVS is continuous.
- Every knot in a bornivorous string is a neighborhood of the origin.
Every
pseudometrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
is quasi-bornological.
A TVS
in which every
bornivorous set
In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology \mathcal is called bornivorous and a bornivore if it absorbs every element of \mathcal.
If X is a topological vector space (TVS) then a ...
is a neighborhood of the origin is a quasi-bornological space.
If
is a quasi-bornological TVS then the finest locally convex topology on
that is coarser than
makes
into a locally convex bornological space.
Bornological space
In functional analysis, a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS)
with a
continuous dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
is called a if it is locally convex and any of the following equivalent conditions holds:
- Every convex, balanced, and bornivorous set in is a neighborhood of zero.
- Every
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
from into a locally convex TVS is continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
.
* Recall that a linear map is bounded if and only if it maps any sequence converging to in the domain to a bounded subset of the codomain. In particular, any linear map that is sequentially continuous at the origin is bounded.
- Every bounded linear operator from into a
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
is continuous.
- Every bounded linear operator from into a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
is continuous.
If
is a
Hausdorff locally convex space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
then we may add to this list:
- The locally convex topology induced by the von Neumann bornology on is the same as 's given topology.
- Every bounded
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
on is continuous.
- Any other Hausdorff locally convex topological vector space topology on that has the same (von Neumann) bornology as is necessarily coarser than
- is the inductive limit of normed spaces.
- is the inductive limit of the normed spaces as varies over the closed and bounded disks of (or as varies over the bounded disks of ).
- carries the Mackey topology and all bounded linear functionals on are continuous.
-
has both of the following properties:
* is or , which means that every convex sequentially open subset of is open,
* is or , which means that every convex and bornivorous subset of is sequentially open.
where a subset of is called if every sequence converging to eventually belongs to
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:
- Any linear map from a locally convex bornological space into a locally convex space that maps null sequences in to bounded subsets of 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
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
is bornological.
* Thus every normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
and Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
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
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
is bornological.
- If the strong dual of a metrizable locally convex space is separable, then it is bornological.
- A vector subspace of a Hausdorff locally convex bornological space that has finite codimension in is bornological.
- The finest locally convex topology on a vector space is bornological.
;Counterexamples
There exists a bornological
LB-space In mathematics, an ''LB''-space, also written (''LB'')-space, is a topological vector space X that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Banach spaces.
This means that X is a direct limit of a direct syste ...
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
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of a locally convex bornological space is complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
.
- Every locally convex bornological space is infrabarrelled.
- Every Hausdorff sequentially complete bornological TVS is ultrabornological.
* Thus every
compete
Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, indivi ...
Hausdorff bornological space is ultrabornological.
* In particular, every Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
is ultrabornological.
- The finite product of locally convex ultrabornological spaces is ultrabornological.
- Every Hausdorff bornological space is quasi-barrelled.
- Given a bornological space with
continuous dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
the topology of coincides with the Mackey topology In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not ma ...
* In particular, bornological spaces are Mackey space
In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space ''X'' such that the topology of ''X'' coincides with the Mackey topology τ(''X'',''X′''), the finest topology which still prese ...
s.
- Every
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
(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).
-
Let be a metrizable locally convex space with continuous dual Then the following are equivalent:
- is bornological.
- is quasi-barrelled.
- is barrelled.
- is a
distinguished space
In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are ...
.
- If is a linear map between locally convex spaces and if is bornological, then the following are equivalent:
- is continuous.
- is sequentially continuous.
- For every set that's bounded in is bounded.
- If is a null sequence in then is a null sequence in
- If is a Mackey convergent null sequence in then is a bounded subset of
- Suppose that and are locally convex TVSs and that the space of continuous linear maps is endowed with the topology of uniform convergence on bounded subsets of If is a bornological space and if is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
then is a complete TVS.
* In particular, the strong dual of a locally convex bornological space is complete. However, it need not be bornological.
;Subsets
- In a locally convex bornological space, every convex bornivorous set is a neighborhood of ( is required to be a disk).
- Every bornivorous subset of a locally convex
metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
is a neighborhood of the origin.
- Closed vector subspaces of bornological space need not be bornological.
Ultrabornological spaces
A disk in a topological vector space
is called if it absorbs all
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
s.
If
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.
- is the inductive limit of the spaces as varies over all compact disks in
- A
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
on that is bounded on each Banach disk is necessarily continuous.
- For every locally convex space and every linear map if is bounded on each Banach disk then is continuous.
- For every Banach space and every linear map if is bounded on each Banach disk then is continuous.
Properties
The finite product of ultrabornological spaces is ultrabornological. Inductive limits of ultrabornological spaces are ultrabornological.
See also
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References
Bibliography
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{{Topological vector spaces
Topological vector spaces