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 ...
, a dual system, dual pair, or 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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s
and
over
and a non-
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
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.
Definition
Vector spaces
Let V, W ...
.
Duality theory, the study of dual systems, is part of
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 space#Definition, inner product, Norm (mathematics)#Defini ...
.
According to
Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of
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 ...
s, for it provides the deepest and most beautiful results of the subject."
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
(which this article assumes 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 form ...
s
) and a bilinear map
which is called the
bilinear map associated with the pairing or simply the pairing's
map/bilinear form.
For every
define
and for every
define
Every
is a linear functional on
and every
is a linear functional on
Let
where each of these sets forms a vector space of linear functionals.
It is common practice to write
instead of
in which case the pair is often denoted by
rather than
However, this article will reserve 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
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
);
#
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
).
In this case say that
is
non-degenerate, say that
places and in duality (or in
separated duality), and
is called the
duality pairing of the
;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 ]
;Orthogonality
The vectors
and
are called
, written
if
Two subsets
and
are
orthogonal, 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''⊥ of all vectors in ''V'' that are orthogonal to every ...
or
annihilator of a subset
is
Polar sets
Throughout,
will be a pairing over
The
absolute polar or
polar of a subset
of
is the set:
Dually, the absolute polar or polar of a subset
of
is denoted by
and defined by
In this case, the absolute polar of a subset
of
is also called the
absolute prepolar or
prepolar of
and may be denoted by
The polar
is necessarily a convex 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
then the
bipolar of
denoted by
is the set
Similarly, if
then the bipolar of
is
If
is a vector subspace of
then
and this is also equal to the
real polar of
Dual definitions and results
Given a pairing
define a new pairing
where
for all
There is a repeating theme in duality theory, which is 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
This conventions also apply to theorems.
:: Adhering to common practice, unless clarity is needed, whenever a definition (or result) for a pairing
is given then this article will omit mention of the corresponding dual definition (or result) but nevertheless use it.
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 it allows us to avoid having to assign a symbol to
:: If a definition and its notation for a pairing
depends on the order of
and
(e.g. the definition of the Mackey topology
on
) then by switching the order of
and
then it is meant that definition applied to
(e.g.
actually denotes the topology
).
For instance, once the weak topology on
is defined, which is denoted by
then this definition will automatically be applied to the pairing
so as to obtain the definition of the weak topology on
where this topology will be denoted by
rather than
;Identification of
with
Although it is technically incorrect and an abuse of notation, this article will also adhere to the following 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 restrictions 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
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 con ...
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
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 cons ...
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 functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
s 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often d ...
is a dual pairing if and only if
is vector space over
or
has dimension
Here it is assumed that the
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
is
conjugate homogeneous
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
in its second coordinate and homogeneous in its first coordinate.
- If is a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
then forms a dual system.
- If is a complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...
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
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other ...
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 In functional analysis and related areas of mathematics, the beta-dual or -dual is a certain linear subspace of the algebraic dual of a sequence space.
Definition
Given a sequence space the -dual of is defined as
:X^:= \left \.
If is an FK- ...
with the bilinear map defined as for forms a dual system.
Weak topology
Suppose that
is a pairing of vector spaces over
If
then the
weak topology on induced by (and
) is the weakest TVS topology on
denoted by
or simply
making all maps
continuous as
ranges over
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 structure but on the algebraic structures of
Similarly, if
then the dual definition of the
weak topology 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 will arise. 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
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
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
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
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
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...
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
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 cons ...
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
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 ...
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
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 ...
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 (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ...
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'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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 t ...
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 staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
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
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, ...
of ; (d) the -closure of ; (e) the -closure of the convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...
of
- The
bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the Polar set, polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions f ...
: 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 set, polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions f ...
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 In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size†...
.
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 staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
in
if and only if
is the
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
* Polar climate, the c ...
of some
-bounded subset of
Transposes
Transpose 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 satisfies:
#
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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
), and
#
where
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 to be confused with the
Hermitian adjoint
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector 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 ...
).
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 for details).
;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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
(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 (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 ...
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
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
(i.e. the topology of pointwise convergence).
Consequently, when the
continuous dual space
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 cons ...
of a
Hausdorff locally convex TVS
is endowed with the
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
, then
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 ...
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
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 con ...
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
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
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
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
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,
;Examples
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 ...
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, a ...
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
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
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 giv ...
) 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
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
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, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
.
# 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, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
, 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^, but lies ...
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
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a pairin ...
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
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
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 nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
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 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 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 ma ...
.
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
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 Ï„(''X'',''X′''), the finest topology which still pres ...
.
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
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 staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
in
if and only if it is a barrel in
The following theorem shows that
barrels
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, u ...
(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
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
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.
Examples
;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
Pair or PAIR or Pairing may refer to:
Government and politics
* Pair (parliamentary convention), matching of members unable to attend, so as not to change the voting margin
* ''Pair'', a member of the Prussian House of Lords
* ''Pair'', the Frenc ...
Topological vector spaces