Dual system


In the field of functional analysis, a subfield of mathematics, a dual system, dual pair, or a duality over a field is a triple consisting of two vector spaces over and a bilinear map such that for all non-zero the map is not identically 0 and for all non-zero, the map is not identically 0.
The study of dual systems is called duality theory.
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 spaces, for it provides the deepest and most beautiful results of the subject."

Definition, notation, and conventions

A pairing or a pair over a field is a triple, which may also be denoted by, consisting of two vector spaces and over and a bilinear map, which we call the bilinear map associated with the pairing or simply the pairing's map/bilinear form.
A pairing is called a dual system, a dual pair, or a duality over if it satisfies the following two separation axioms:
  1. separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero, the map is not identically 0 ;
  2. separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero, the map is not identically 0.
In this case we say that is non-degenerate, we say that places and in duality, and we call the duality pairing of the.
A subset of is called total if for all, for all implies.
A total subset of is defined analogously.
We say that the elements and are orthogonal and write if.
We say that two sets and are orthogonal and write if r and s are orthogonal for all and and we say that is orthogonal to an element if is orthogonal to.
For, we define the orthogonal or annihilator of to be.

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 ⊆ Y 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 we can define a new pairing where for all and all.
For the purpose of explaining the following convention, we will call the mirror of .
There is a repeating theme in duality theory, which is that any definition for a pairing has a corresponding dual definition for the pairing.
For instance, if we define " distinguishes points of " as above, then under this convention we immediately obtain the dual definition of " distinguishes points of ".
This following notation is almost ubiquitous because it allows us to avoid having to assign a symbol to the mirror of.
For instance, once we define the weak topology on, which is denoted by, then we will automatically apply this definition to the pairing so as to obtain the definition of the weak topology on, where we will denote the this topology rather than.
;Identification of with
Although it is technically incorrect and an abuse of notation, we will also adhere to the following nearly ubiquitous convention:

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.
Note that if is a duality then it's possible for a restrictions to fail to be a duality.

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 we will instead call it 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.
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.

Other examples

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 all maps continuous, as ranges over.
We use,, or simply to denote endowed with the weak topology.
If we do not indicate what the subset is, then by the weak topology on we mean the weak topology on induced by.
Similarly, if then we have the dual definition of the weak topology on induced by , which is denoted by or simply .
Observe that the weak topology depends entirely on the function and the usual topology on .
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 we say that -converges to x if converges to in.
A net -converges to x if and only if for all, converges to.
Note that 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 for all, where.
;Hausdorffness

Weak representation theorem

The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of.

Orthogonals, quotients, and subspaces

If is a pairing then for any subset of :
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

The following results are important for defining polar topologies.
The bipolar theorem in particular "is an indispensable tool in working with dualities."

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.
We will say that s transpose or adjoint is well-defined if the following conditions are satisfies:
  1. distinguishes points of , and
  2. , where.
In this case, for any there exists a unique, where we will denote by this element of by.
This defines a linear map
called the transpose of 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.
Note that for every, the defining condition for is
that is,
Note that 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.
If and are normed spaces under their canonical dualities and if is a continuous linear map, then.

Weak continuity

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

We call -complete or weakly-complete if is a complete vector space.
Note that there exist Banach spaces that are not weakly-complete.
In particular, if is a vector subspace of such that separates points of, 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 we can assume 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 spacial 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 F and it will be denoted by ;
that is,.
In this case, for all,
where the defining condition for is:
or equivalently, for all.
;Examples
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 ℰ of.

Weak continuity and openness

Transpose of a map between TVSs
The transpose of map between two TVSs is defined if and only if is weakly continuous.

Metrizability and separability

Polar topologies and topologies compatible with pairing

Starting with only the weak topology, we may obtain a range of locally convex topologies by using polar sets.
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

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?.
Observe that 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
;Bounded subsets

Topologies compatible with a pair

If is a pairing over and is a vector topology on then we say that is a topology of the pair and that it is compatible with the pair 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:.
Note that 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.
Note that if is a normed space that is not reflexive then the usual norm topology on its continuous dual space is not 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 real linear functional on.
Note that the above theorem implies that the closed and convex subsets of a locally convex space depend entirely 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.
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 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.