In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a dual system, dual pair or a duality over a
field is a triple
consisting of two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s,
and
, over
and a non-
degenerate bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
A bilinear map can also be defined for ...
.
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
,
duality is the study of dual systems and is important 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. Duality plays crucial roles in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
because it has extensive applications to the theory of
Hilbert spaces
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
.
Definition, notation, and conventions
Pairings
A
or
pair over a field
is a triple
which may also be denoted by
consisting of two vector spaces
and
over
and a
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
A bilinear map can also be defined for ...
called the
bilinear map associated with the pairing, or more simply called the pairing's
map or its
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
. The examples here only describe when
is either the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s or the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
, but the mathematical theory is general.
For every
, define
and for every
define
Every
is a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
on
and every
is a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
on
. Therefore both
form vector spaces of
linear functionals
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mat ...
.
It is common practice to write
instead of
, in which in some cases the pairing may be denoted by
rather than
. However, this article will reserve the use of
for the canonical
evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.
Dual pairings
A pairing
is called a , a , or a over
if the
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
is non-
degenerate, which means that it satisfies the following two separation axioms:
#
separates (distinguishes) points of : if
is such that
then
; or equivalently, for all non-zero
, the map
is not identically
(i.e. there exists a
such that
for each
);
#
separates (distinguishes) points of : if
is such that
then
; or equivalently, for all non-zero
the map
is not identically
(i.e. there exists an
such that
for each
).
In this case
is
non-degenerate, and one can say that
places and in duality (or, redundantly but explicitly, in
separated duality), and
is called the
duality pairing of the triple
.
Total subsets
A subset
of
is called
if for every
,
implies
A total subset of
is defined analogously (see footnote).
[A subset of is total if for all , implies .] Thus
separates points of
if and only if
is a total subset of
, and similarly for
.
Orthogonality
The vectors
and
are
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
, written
, if
. Two subsets
and
are
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
, written
, if
; that is, if
for all
and
. The definition of a subset being orthogonal to a vector is defined
analogously.
The
orthogonal complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^\perp of all vectors in V that are orthogonal to every vector in W. I ...
or
annihilator of a subset
is
Thus
is a total subset of
if and only if
equals
.
Polar sets
Given a triple
defining a pairing over
, the
absolute polar set or
polar set of a subset
of
is the set:
Symmetrically, the absolute polar set or polar set of a subset
of
is denoted by
and defined by
To use bookkeeping that helps keep track of the anti-symmetry of the two sides of the duality, the absolute polar of a subset
of
may also be called the
absolute prepolar or
prepolar of
and then may be denoted by
.
The polar
is necessarily a
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 ...
set containing
where if
is balanced then so is
and if
is a vector subspace of
then so too is
a vector subspace of
If
is a vector subspace of
then
and this is also equal to the
real polar of
If
then the
bipolar of
, denoted
, is the polar of the orthogonal complement of
, i.e., the set
Similarly, if
then the bipolar of
is
Dual definitions and results
Given a pairing
define a new pairing
where
for all
and
.
There is a consistent theme in duality theory that any definition for a pairing
has a corresponding dual definition for the pairing
:: Given any definition for a pairing
one obtains a by applying it to the pairing
These conventions also apply to theorems.
For instance, if "
distinguishes points of
" (resp, "
is a total subset of
") is defined as above, then this convention immediately produces the dual definition of "
distinguishes points of
" (resp, "
is a total subset of
").
This following notation is almost ubiquitous and allows us to avoid assigning a symbol to
:: If a definition and its notation for a pairing
depends on the order of
and
(for example, the definition of 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 ...
on
) then by switching the order of
and
then it is meant that definition applied to
(continuing the same example, the topology
would actually denote the topology
).
For another example, once the weak topology on
is defined, denoted by
, then this dual definition would automatically be applied to the pairing
so as to obtain the definition of the weak topology on
, and this topology would be denoted by
rather than
.
Identification of with
Although it is technically incorrect and an abuse of notation, this article will adhere to the nearly ubiquitous convention of treating a pairing
interchangeably with
and also of denoting
by
Examples
Restriction of a pairing
Suppose that
is a pairing,
is a vector subspace of
and
is a vector subspace of
. Then the
restriction of
to
is the pairing
If
is a duality, then it's possible for a restriction to fail to be a duality (e.g. if
and
).
This article will use the common practice of denoting the restriction
by
Canonical duality on a vector space
Suppose that
is a vector space and let
denote the
algebraic dual space of
(that is, the space of all linear functionals on
).
There is a canonical duality
where
which is called the
evaluation map or the
natural or
canonical bilinear functional on
Note in particular that for any
is just another way of denoting
; i.e.
If
is a vector subspace of
, then the restriction of
to
is called the
canonical pairing where if this pairing is a duality then it is instead called the
canonical duality. Clearly,
always distinguishes points of
, so the canonical pairing is a dual system if and only if
separates points of
The following notation is now nearly ubiquitous in duality theory.
The evaluation map will be denoted by
(rather than by
) and
will be written rather than
:Assumption: As is common practice, if
is a vector space and
is a vector space of linear functionals on
then unless stated otherwise, it will be assumed that they are associated with the canonical pairing
If
is a vector subspace of
then
distinguishes points of
(or equivalently,
is a duality) if and only if
distinguishes points of
or equivalently if
is total (that is,
for all
implies
).
Canonical duality on a topological vector space
Suppose
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) with
continuous dual space
Then the restriction of the canonical duality
to
×
defines a pairing
for which
separates points of
If
separates points of
(which is true if, for instance,
is a Hausdorff locally convex space) then this pairing forms a duality.
:Assumption: As is commonly done, whenever
is a TVS, then unless indicated otherwise, it will be assumed without comment that it's associated with the canonical pairing
Polars and duals of TVSs
The following result shows that the
continuous linear functionals on a TVS are exactly those linear functionals that are bounded on a neighborhood of the origin.
Inner product spaces and complex conjugate spaces
A
pre-Hilbert space is a dual pairing if and only if
is vector space over
or
has dimension
Here it is assumed that the
sesquilinear form is
conjugate homogeneous in its second coordinate and homogeneous in its first coordinate.
- If is a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
then forms a dual system.
- If is a complex
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
then forms a dual system if and only if If is non-trivial then does not even form pairing since the inner product is sesquilinear rather than bilinear.
Suppose that
is a complex pre-Hilbert space with scalar multiplication denoted as usual by juxtaposition or by a dot
Define the map
where the right-hand side uses the scalar multiplication of
Let
denote the
complex conjugate vector space of
where
denotes the additive group of
(so vector addition in
is identical to vector addition in
) but with scalar multiplication in
being the map
(instead of the scalar multiplication that
is endowed with).
The map
defined by
is linear in both coordinates
[That is linear in its first coordinate is obvious. Suppose is a scalar. Then which shows that is linear in its second coordinate.] and so
forms a dual pairing.
Other examples
- Suppose and for all let Then is a pairing such that distinguishes points of but does not distinguish points of Furthermore,
- Let (where is such that ), and Then is a dual system.
- Let and be vector spaces over the same field Then the bilinear form places and in duality.
- A
sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
and its beta dual with the bilinear map defined as for forms a dual system.
Weak topology
Suppose that
is a pairing of
vector spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and sc ...
over
If
then the
weak topology on induced by (and
) is the weakest TVS topology on
denoted by
or simply
making each map
continuous as a function of
for every
. If
is not clear from context then it should be assumed to be all of
in which case it is called the
weak topology on
(induced by
).
The notation
or (if no confusion could arise) simply
is used to denote
endowed with the weak topology
Importantly, the weak topology depends on the function
the usual topology on
and
's
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
structure but on the
algebraic structures
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
of
Similarly, if
then the dual definition of the
weak topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on induced by (and
), which is denoted by
or simply
(see footnote for details).
[The weak topology on is the weakest TVS topology on making all maps continuous, as ranges over The dual notation of or simply may also be used to denote endowed with the weak topology If is not clear from context then it should be assumed to be all of in which case it is simply called the weak topology on (induced by ).]
:: If "
" is attached to a topological definition (e.g.
-converges,
-bounded,
etc.) then it means that definition when the first space (i.e.
) carries the
topology. Mention of
or even
and
may be omitted if no confusion arises. So, for instance, if a sequence
in
"
-converges" or "weakly converges" then this means that it converges in
whereas if it were a sequence in
, then this would mean that it converges in
).
The topology
is
locally convex since it is determined by the family of seminorms
defined by
as
ranges over
If
and
is a
net in
then
-converges to
if
converges to
in
A net
-converges to
if and only if for all
converges to
If
is a sequence of
orthonormal vectors in Hilbert space, then
converges weakly to 0 but does not norm-converge to 0 (or any other vector).
If
is a pairing and
is a proper vector subspace of
such that
is a dual pair, then
is strictly
coarser than
Bounded subsets
A subset
of
is
-bounded if and only if
where
Hausdorffness
If
is a pairing then the following are equivalent:
#
distinguishes points of
;
# The map
defines an
injection from
into the algebraic dual space of
;
#
is
Hausdorff.
Weak representation theorem
The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of
Consequently, the
continuous dual space of
is
With respect to the canonical pairing, if
is a TVS whose continuous dual space
separates points on
(i.e. such that
is Hausdorff, which implies that
is also necessarily Hausdorff) then the continuous dual space of
is equal to the set of all "evaluation at a point
" maps as
ranges over
(i.e. the map that send
to
).
This is commonly written as
This very important fact is why results for polar topologies on continuous dual spaces, such as the
strong dual topology on
for example, can also often be applied to the original TVS
; for instance,
being identified with
means that the topology
on
can instead be thought of as a topology on
Moreover, if
is endowed with a topology that is
finer than
then the continuous dual space of
will necessarily contain
as a subset.
So for instance, when
is endowed with the strong dual topology (and so is denoted by
) then
which (among other things) allows for
to be endowed with the subspace topology induced on it by, say, the strong dual topology
(this topology is also called the strong
bidual topology and it appears in the theory of
reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
s: the Hausdorff locally convex TVS
is said to be if
and it will be called if in addition the strong bidual topology
on
is equal to
's original/starting topology).
Orthogonals, quotients, and subspaces
If
is a pairing then for any subset
of
:
- and this set is -closed;
- ;
* Thus if is a -closed vector subspace of then
- If is a family of -closed vector subspaces of then
- If is a family of subsets of then
If
is a normed space then under the canonical duality,
is norm closed in
and
is norm closed in
Subspaces
Suppose that
is a vector subspace of
and let
denote the restriction of
to
The weak topology
on
is identical to the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
that
inherits from
Also,
is a paired space (where
means
) where
is defined by
The topology
is equal to the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
that
inherits from
Furthermore, if
is a dual system then so is
Quotients
Suppose that
is a vector subspace of
Then
is a paired space where
is defined by
The topology
is identical to the usual
quotient topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
induced by
on
Polars and the weak topology
If
is a locally convex space and if
is a subset of the continuous dual space
then
is
-bounded if and only if
for some
barrel
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
in
The following results are important for defining polar topologies.
If
is a pairing and
then:
- The polar of is a closed subset of
- The polars of the following sets are identical: (a) ; (b) the convex hull of ; (c) the balanced hull of ; (d) the -closure of ; (e) the -closure of the convex balanced hull of
- The
bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone ...
: The bipolar of denoted by is equal to the -closure of the convex balanced hull of
* The bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone ...
in particular "is an indispensable tool in working with dualities."
- is -bounded if and only if is absorbing in
- If in addition distinguishes points of then is - bounded if and only if it is - totally bounded.
If
is a pairing and
is a locally convex topology on
that is consistent with duality, then a subset
of
is a
barrel
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
in
if and only if
is the
polar of some
-bounded subset of
Transposes
Transposes of a linear map with respect to pairings
Let
and
be pairings over
and let
be a linear map.
For all
let
be the map defined by
It is said that
s
transpose or
adjoint is well-defined if the following conditions are satisfied:
#
distinguishes points of
(or equivalently, the map
from
into the algebraic dual
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
), and
#
where
and
.
In this case, for any
there exists (by condition 2) a unique (by condition 1)
such that
), where this element of
will be denoted by
This defines a linear map
called the
transpose or
adjoint of with respect to and (this should not be confused with the
Hermitian adjoint
In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where \l ...
).
It is easy to see that the two conditions mentioned above (i.e. for "the transpose is well-defined") are also necessary for
to be well-defined.
For every
the defining condition for
is
that is,
for all
By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form
[If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is:
]
[If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is: ]
[If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is: ]
[If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is: ] etc. (see footnote).
Properties of the transpose
Throughout,
and
be pairings over
and
will be a linear map whose transpose
is well-defined.
*
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
(i.e.
) if and only if the range of
is dense in
* If in addition to
being well-defined, the transpose of
is also well-defined then
* Suppose
is a pairing over
and
is a linear map whose transpose
is well-defined. Then the transpose of
which is
is well-defined and
* If
is a vector space isomorphism then
is bijective, the transpose of
which is
is well-defined, and
* Let
and let
denotes the
absolute polar of
then:
*#
;
*# if
for some
then
;
*# if
is such that
then
;
*# if
and
are weakly closed disks then
if and only if
;
*#
: These results hold when the
real polar is used in place of the absolute polar.
If
and
are normed spaces under their canonical dualities and if
is a continuous linear map, then
Weak continuity
A linear map
is
weakly continuous (with respect to
and
) if
is continuous.
The following result shows that the existence of the transpose map is intimately tied to the weak topology.
Weak topology and the canonical duality
Suppose that
is a vector space and that
is its the algebraic dual.
Then every
-bounded subset of
is contained in a finite dimensional vector subspace and every vector subspace of
is
-closed.
Weak completeness
If
is a
complete topological vector space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
say that
is
-complete or (if no ambiguity can arise)
weakly-complete.
There exist
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
s that are not weakly-complete (despite being complete in their norm topology).
If
is a vector space then under the canonical duality,
is complete.
Conversely, if
is a Hausdorff
locally convex TVS with continuous dual space
then
is complete if and only if
; that is, if and only if the map
defined by sending
to the evaluation map at
(i.e.
) is a bijection.
In particular, with respect to the canonical duality, if
is a vector subspace of
such that
separates points of
then
is complete if and only if
Said differently, there does exist a proper vector subspace
of
such that
is Hausdorff and
is complete in the
weak-* topology (i.e. the topology of pointwise convergence).
Consequently, when the
continuous dual space of a
Hausdorff locally convex TVS
is endowed with the
weak-* topology, then
is
complete if and only if
(that is, if and only if linear functional on
is continuous).
Identification of ''Y'' with a subspace of the algebraic dual
If
distinguishes points of
and if
denotes the range of the injection
then
is a vector subspace of the
algebraic dual space of
and the pairing
becomes canonically identified with the canonical pairing
(where
is the natural evaluation map).
In particular, in this situation it will be assumed
without loss of generality that
is a vector subspace of
's algebraic dual and
is the evaluation map.
:: Often, whenever
is injective (especially when
forms a dual pair) then it is common practice to assume
without loss of generality that
is a vector subspace of the algebraic dual space of
that
is the natural evaluation map, and also denote
by
In a completely analogous manner, if
distinguishes points of
then it is possible for
to be identified as a vector subspace of
's algebraic dual space.
Algebraic adjoint
In the special case where the dualities are the canonical dualities
and
the transpose of a linear map
is always well-defined.
This transpose is called the
algebraic adjoint of
and it will be denoted by
;
that is,
In this case, for all
where the defining condition for
is:
or equivalently,
If
for some integer
is a basis for
with
dual basis
In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimension of V), the dual set of B is a set B^* of vectors in the dual space V^* with the same index set I such that B and ...
is a linear operator, and the matrix representation of
with respect to
is
then the transpose of
is the matrix representation with respect to
of
Weak continuity and openness
Suppose that
and
are canonical pairings (so
and
) that are dual systems and let
be a linear map.
Then
is weakly continuous if and only if it satisfies any of the following equivalent conditions:
#
is continuous.
#
# the transpose of ''F'',
with respect to
and
is well-defined.
If
is weakly continuous then
will be continuous and furthermore,
A map
between topological spaces is
relatively open if
is an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
ping, where
is the range of
Suppose that
and
are dual systems and
is a weakly continuous linear map.
Then the following are equivalent:
#
is relatively open.
# The range of
is
-closed in
;
#
Furthermore,
*
is injective (resp. bijective) if and only if
is surjective (resp. bijective);
*
is surjective if and only if
is relatively open and injective.
=Transpose of a map between TVSs
=
The transpose of map between two TVSs is defined if and only if
is weakly continuous.
If
is a linear map between two Hausdorff locally convex topological vector spaces, then:
* If
is continuous then it is weakly continuous and
is both Mackey continuous and strongly continuous.
* If
is weakly continuous then it is both Mackey continuous and strongly continuous (defined below).
* If
is weakly continuous then it is continuous if and only if
maps
equicontinuous subsets of
to equicontinuous subsets of
* If
and
are normed spaces then
is continuous if and only if it is weakly continuous, in which case
* If
is continuous then
is relatively open if and only if
is weakly relatively open (i.e.
is relatively open) and every equicontinuous subsets of
is the image of some equicontinuous subsets of
* If
is continuous injection then
is a TVS-embedding (or equivalently, a
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
) if and only if every equicontinuous subsets of
is the image of some equicontinuous subsets of
Metrizability and separability
Let
be a
locally convex space with continuous dual space
and let
# If
is
equicontinuous or
-compact, and if
is such that
is dense in
then the subspace topology that
inherits from
is identical to the subspace topology that
inherits from
# If
is
separable and
is equicontinuous then
when endowed with the subspace topology induced by
is
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
.
# If
is separable and
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
, then
is separable.
# If
is a normed space then
is separable if and only if the closed unit call the continuous dual space of
is metrizable when given the subspace topology induced by
# If
is a normed space whose continuous dual space is separable (when given the usual norm topology), then
is separable.
Polar topologies and topologies compatible with pairing
Starting with only the weak topology, the use of
polar set
In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X, lying in the dual space X^.
The bipolar of a subset is the polar of A^\circ, but ...
s produces a range of locally convex topologies.
Such topologies are called
polar topologies.
The weak topology is the
weakest topology of this range.
Throughout,
will be a pairing over
and
will be a non-empty collection of
-bounded subsets of
Polar topologies
Given a collection
of subsets of
, the
polar topology on
determined by
(and
) or the
-topology on
is the unique
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) topology on
for which
forms a
subbasis of neighborhoods at the origin.
When
is endowed with this
-topology then it is denoted by ''Y''
.
Every polar topology is necessarily
locally convex.
When
is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
with respect to subset inclusion (i.e. if for all
there exists some
such that
) then this neighborhood subbasis at 0 actually forms a
neighborhood basis at 0.
The following table lists some of the more important polar topologies.
:: If
denotes a polar topology on
then
endowed with this topology will be denoted by
or simply
(e.g. for
we'd have
so that
and
all denote
endowed with
).
Definitions involving polar topologies
Continuity
A linear map
is
Mackey continuous (with respect to
and
) if
is continuous.
A linear map
is
strongly continuous (with respect to
and
) if
is continuous.
Bounded subsets
A subset of
is
weakly bounded (resp.
Mackey bounded,
strongly bounded) if it is bounded in
(resp. bounded in
bounded in
).
Topologies compatible with a pair
If
is a pairing over
and
is a vector topology on
then
is a
topology of the pairing and that it is
compatible (or
consistent)
with the pairing if it is
locally convex and if the continuous dual space of
[Of course, there is an analogous definition for topologies on to be "compatible it a pairing" but this article will only deal with topologies on ]
If
distinguishes points of
then by identifying
as a vector subspace of
's algebraic dual, the defining condition becomes:
Some authors (e.g.
rèves 2006and
chaefer 1999 require that a topology of a pair also be Hausdorff, which it would have to be if
distinguishes the points of
(which these authors assume).
The weak topology
is compatible with the pairing
(as was shown in the Weak representation theorem) and it is in fact the weakest such topology.
There is a strongest topology compatible with this pairing and that is 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 ...
.
If
is a normed space that is not
reflexive then the usual norm topology on its continuous dual space is compatible with the duality
Mackey–Arens theorem
The following is one of the most important theorems in duality theory.
It follows that the Mackey topology
which recall is the polar topology generated by all
-compact disks in
is the strongest locally convex topology on
that is compatible with the pairing
A locally convex space whose given topology is identical to the Mackey topology is called a
Mackey space.
The following consequence of the above Mackey-Arens theorem is also called the Mackey-Arens theorem.
Mackey's theorem, barrels, and closed convex sets
If
is a TVS (over
or
) then a
half-space is a set of the form
for some real
and some continuous linear functional
on
The above theorem implies that the closed and convex subsets of a locally convex space depend on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality;that is, if
and
are any locally convex topologies on
with the same continuous dual spaces, then a convex subset of
is closed in the
topology if and only if it is closed in the
topology.
This implies that the
-closure of any convex subset of
is equal to its
-closure and that for any
-closed
disk in
In particular, if
is a subset of
then
is a
barrel
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
in
if and only if it is a barrel in
The following theorem shows that
barrels (i.e. closed
absorbing disks) are exactly the polars of weakly bounded subsets.
If
is a topological vector space, then:
# A closed
absorbing and
balanced subset
of
absorbs each convex compact subset of
(i.e. there exists a real
such that
contains that set).
# If
is Hausdorff and locally convex then every barrel in
absorbs every convex bounded complete subset of
All of this leads to Mackey's theorem, which is one of the central theorems in the theory of dual systems.
In short, it states the bounded subsets are the same for any two Hausdorff locally convex topologies that are compatible with the same duality.
Space of finite sequences
Let
denote the space of all sequences of scalars
such that
for all sufficiently large
Let
and define a bilinear map
by
Then
Moreover, a subset
is
-bounded (resp.
-bounded) if and only if there exists a sequence
of positive real numbers such that
for all
and all indices
(resp. and
).
It follows that there are weakly bounded (that is,
-bounded) subsets of
that are not strongly bounded (that is, not
-bounded).
See also
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Notes
References
Bibliography
*
* Michael Reed and Barry Simon, ''Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis,'' Section III.3. Academic Press, San Diego, 1980. .
*
*
*
*
External links
Duality Theory
{{DEFAULTSORT:Dual Pair
Functional analysis
Pair
Linear functionals
Topological vector spaces