Dual system


In mathematics, a dual system, dual pair or a duality over a field is a triple consisting of two vector spaces, and, over and a non-degenerate bilinear map.
In mathematics, duality is the study of dual systems and is important in functional analysis. Duality plays crucial roles in quantum mechanics because it has extensive applications to the theory of Hilbert spaces.

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 called the bilinear map associated with the pairing, or more simply called the pairing's map or its bilinear form. The examples here only describe when is either the real numbers or the complex numbers, but the mathematical theory is general.
For every, define
and for every define
Every is a linear functional on and every is a linear functional on. Therefore both
form vector spaces of linear functionals.
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 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:
  1. separates points of : if is such that then ; or equivalently, for all non-zero, the map is not identically ;
  2. separates points of : if is such that then ; or equivalently, for all non-zero the map is not identically .
In this case is non-degenerate, and one can say that places and in 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. Thus separates points of if and only if is a total subset of, and similarly for.

Orthogonality

The vectors and are orthogonal, 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 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 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
For instance, if " distinguishes points of " is defined as above, then this convention immediately produces the dual definition of " distinguishes points of ".
This following notation is almost ubiquitous and allows us to avoid assigning a symbol to
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 (X, Y) with (Y, X)

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.
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 .
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 and will be written rather than
If is a vector subspace of then distinguishes points of if and only if distinguishes points of or equivalently if is total.

Canonical duality on a topological vector space

Suppose is a topological vector space with continuous dual space
Then the restriction of the canonical duality to × defines a pairing for which separates points of
If separates points of then this pairing forms a duality.

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 then forms a dual system.
  • If is a complex Hilbert space 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 but with scalar multiplication in being the map .
The map defined by is linear in both coordinates 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, 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 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 over
If then the weak topology on induced by 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 .
The notation or 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 , which is denoted by or simply .
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.
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:
  1. distinguishes points of ;
  2. The map defines an injection from into the algebraic dual space of ;
  3. 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 then the continuous dual space of is equal to the set of all "evaluation at a point " maps as ranges over .
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 then
which allows for to be endowed with the subspace topology induced on it by, say, the strong dual 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 that inherits from
Also, is a paired space where is defined by
The topology is equal to the subspace 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 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 in
The following results are important for defining polar topologies.
If is a pairing and then:

  1. The polar of is a closed subset of
  2. The polars of the following sets are identical: ; the convex hull of ; the balanced hull of ; the -closure of ; the -closure of the convex balanced hull of
  3. The bipolar theorem: The bipolar of denoted by is equal to the -closure of the convex balanced hull of
    • The bipolar theorem in particular "is an indispensable tool in working with dualities."
  4. is -bounded if and only if is absorbing in
  5. 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 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:
  1. distinguishes points of, and
  2. where and.
In this case, for any there exists a unique, where this element of will be denoted by
This defines a linear map
called the transpose or adjoint of with respect to and .
It is easy to see that the two conditions mentioned above 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
etc..

Properties of the transpose

Throughout, and be pairings over and will be a linear map whose transpose is well-defined.
  • is injective 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 ;
  • #
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 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 say that is -complete or weakly-complete.
There exist Banach spaces that are not weakly-complete.
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 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.
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 .

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 .
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.
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 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 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:
  1. is continuous.
  3. 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 mapping, where is the range of
Suppose that and are dual systems and is a weakly continuous linear map.
Then the following are equivalent:
  1. is relatively open.
  2. The range of is -closed in ;
Furthermore,
  • is injective if and only if is surjective ;
  • 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.
  • 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 and every equicontinuous subsets of is the image of some equicontinuous subsets of
  • If is continuous injection then is a TVS-embedding 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
  1. 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
  2. If is separable and is equicontinuous then when endowed with the subspace topology induced by is metrizable.
  3. If is separable and metrizable, then is separable.
  4. 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
  5. If is a normed space whose continuous dual space is separable, then is separable.

Polar topologies and topologies compatible with pairing

Starting with only the weak topology, the use of polar sets 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 or the -topology on is the unique topological vector space 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 with respect to subset inclusion then this neighborhood subbasis at 0 actually forms a neighborhood basis at 0.
The following table lists some of the more important polar topologies.

NotationName Alternative name
finite subsets of

pointwise/simple convergenceweak/weak* topology
-compact disksMackey topology
-compact convex subsetscompact convex convergence
-compact subsets
compact convergence
-bounded subsets
bounded convergencestrong topology
Strongest polar topology

Definitions involving polar topologies

Continuity
A linear map is Mackey continuous if is continuous.
A linear map is strongly continuous if is continuous.

Bounded subsets

A subset of is weakly bounded if it is 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 with the pairing if it is locally convex and if the continuous dual space of
If distinguishes points of then by identifying as a vector subspace of 's algebraic dual, the defining condition becomes:
Some authors require that a topology of a pair also be Hausdorff, which it would have to be if distinguishes the points of .
The weak topology is compatible with the pairing and it is in fact the weakest such topology.
There is a strongest topology compatible with this pairing and that is the Mackey topology.
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 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 in if and only if it is a barrel in
The following theorem shows that barrels are exactly the polars of weakly bounded subsets.
If is a topological vector space, then:
  1. A closed absorbing and balanced subset of absorbs each convex compact subset of .
  2. 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 if and only if there exists a sequence of positive real numbers such that for all and all indices .
It follows that there are weakly bounded subsets of that are not strongly bounded.