Polar topology

In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.

Preliminaries

A pairing is a triple consisting of two vector spaces over a field and a bilinear map
A dual pair or dual system is a pairing satisfying the following two separation axioms:

separates/distinguishes points of : for all non-zero there exists such that and
separates/distinguishes points of : for all non-zero there exists such that

Polars

The polar or absolute polar of a subset is the set
Dually, the polar or absolute polar of a subset is denoted by and defined by
In this case, the absolute polar of a subset is also called the prepolar of and may be denoted by
The polar is a convex balanced set containing the origin.
If then the bipolar of denoted by is defined by Similarly, if then the bipolar of is defined to be

Weak topologies

Suppose that is a pairing of vector spaces over
The weak topology on induced by is the weakest TVS topology on denoted by or simply making all maps continuous, as ranges over Similarly, there are the dual definition of the weak topology on induced by, which is denoted by or simply : it is the weakest TVS topology on making all maps continuous, as ranges over

Weak boundedness and absorbing polars

It is because of the following theorem that it is almost always assumed that the family consists of -bounded subsets of

Dual definitions and results

Every pairing can be associated with a corresponding pairing where by definition
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, after defining " distinguishes points of " as above, then the dual definition of " distinguishes points of " is immediately obtained.
For instance, once is defined then it should be automatically assume that has been defined without mentioning the analogous definition.
The same applies to many theorems.
In particular, although this article will only define the general notion of polar topologies on with being a collection of -bounded subsets of this article will nevertheless use the dual definition for polar topologies on with being a collection of -bounded subsets of
;Identification of with
Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous:

Polar topologies

Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of
For every and is convex and balanced and because is a -bounded, the set is absorbing in
The polar topology on determined by, also called the -topology on or the topology of uniform convergence on the sets of is the unique topological vector space topology on for which
forms a neighbourhood subbasis at the origin. When is endowed with this -topology then it is denoted by
If is a sequence of positive numbers converging to then the defining neighborhood subbasis at may be replaced with
without changing the resulting topology.
When is a directed set with respect to subset inclusion then the defining neighborhood subbasis at the origin actually forms a neighborhood basis at
;Seminorms defining the polar topology
Every determines a seminorm defined by
where and is in fact the Minkowski functional of Because of this, the -topology on is always a locally convex topology.
;Modifying
If every positive scalar multiple of a set in is contained in some set belonging to then the defining neighborhood subbasis at the origin can be replaced with
without changing the resulting topology.
The following theorem gives ways in which can be modified without changing the resulting -topology on
It is because of this theorem that many authors often require that also satisfy the following additional conditions:

The union of any two sets is contained in some set ;
All scalar multiples of every belongs to

Some authors further assume that every belongs to some set because this assumption suffices to ensure that the -topology is Hausdorff.
;Convergence of nets and filters
If is a net in then in the -topology on if and only if for every or in words, if and only if for every the net of linear functionals on converges uniformly to on ; here, for each the linear functional is defined by
If then in the -topology on if and only if for all
A filter on converges to an element in the -topology on if converges uniformly to on each

Properties

Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of
;Hausdorffness
Proof of :
If then we're done, so assume otherwise. Since the -topology on is a TVS topology, it suffices to show that the set is closed in Let be non-zero, let be defined by for all and let
Since distinguishes points of there exists some such that where it can be assumed without loss of generality that The set is a -open subset of that is not empty. Since is a -dense subset of there exists some and some such that Since so that where is a subbasic closed neighborhood of the origin in the -topology on ■

Examples of polar topologies induced by a pairing

Throughout, will be a pairing of vector spaces over the field and will be a non-empty collection of -bounded subsets of
The following table will omit mention of The topologies are listed in an order that roughly corresponds with coarser topologies first and the finer topologies last; note that some of these topologies may be out of order e.g. and the topology below it or if is not Hausdorff. If more than one collection of subsets appears the same row in the left-most column then that means that the same polar topology is generated by these collections.

	Notation	Name	Alternative name
finite subsets of		pointwise/simple convergence	weak/weak* topology
-compact disks			Mackey topology
-compact convex subsets		compact convex convergence
-compact subsets		compact convergence
-complete and bounded disks		convex balanced complete bounded convergence
-precompact/totally bounded subsets		precompact convergence
-infracomplete and bounded disks		convex balanced infracomplete bounded convergence
-bounded subsets		bounded convergence	strong topology Strongest polar topology

Weak topology σ(''Y'', ''X'')

For any a basic -neighborhood of in is a set of the form:
for some real and some finite set of points in
The continuous dual space of is where more precisely, this means that a linear functional on belongs to this continuous dual space if and only if there exists some such that for all The weak topology is the coarsest TVS topology on for which this is true.
In general, the convex balanced hull of a -compact subset of need not be -compact.
If and are vector spaces over the complex numbers then let and denote these spaces when they are considered as vector spaces over the real numbers Let denote the real part of and observe that is a pairing. The weak topology on is identical to the weak topology This ultimately stems from the fact that for any complex-valued linear functional on with real part then

Mackey topology τ(''Y'', ''X'')

The continuous dual space of is . Moreover, the Mackey topology is the finest locally convex topology on for which this is true, which is what makes this topology important.
Since in general, the convex balanced hull of a -compact subset of need not be -compact, the Mackey topology may be strictly coarser than the topology Since every -compact set is -bounded, the Mackey topology is coarser than the strong topology

Strong topology ?(''Y'', ''X'')

A neighborhood basis at the origin for the topology is:
The strong topology is finer than the Mackey topology.

Polar topologies and topological vector spaces

Throughout this section, will be a topological vector space with continuous dual space and will be the canonical pairing, where by definition The vector space always distinguishes/separates the points of but may fail to distinguishes the points of, in which case the pairing is not a dual pair. By the Hahn–Banach theorem, if is a Hausdorff locally convex space then separates points of and thus forms a dual pair.

Properties

If covers then the canonical map from into is well-defined. That is, for all the evaluation functional on meaning the map is continuous on
- If in addition separates points on then the canonical map of into is an injection.
Suppose that is a continuous linear and that and are collections of bounded subsets of and respectively, that each satisfy axioms and Then the transpose of is continuous if for every there is some such that
- In particular, the transpose of is continuous if carries the topology and carry any topology stronger than the topology.
If is a locally convex Hausdorff TVS over the field and is a collection of bounded subsets of that satisfies axioms and then the bilinear map defined by is continuous if and only if is normable and the -topology on is the strong dual topology
Suppose that is a Fréchet space and is a collection of bounded subsets of that satisfies axioms and If contains all compact subsets of then is complete.

Polar topologies on the continuous dual space

Throughout, will be a TVS over the field with continuous dual space and and will be associated with the canonical pairing. The table below defines many of the most common polar topologies on

	Notation	Name	Alternative name
finite subsets of		pointwise/simple convergence	weak/weak* topology
compact convex subsets		compact convex convergence
compact subsets		compact convergence
-compact disks			Mackey topology
precompact/totally bounded subsets		precompact convergence
complete and bounded disks		convex balanced complete bounded convergence
infracomplete and bounded disks		convex balanced infracomplete bounded convergence
bounded subsets		bounded convergence	strong topology
-compact disks in			Mackey topology

The reason why some of the above collections induce the same polar topologies is due to some basic results. A closed subset of a complete TVS is complete and that a complete subset of a Hausdorff and complete TVS is closed. Furthermore, in every TVS, compact subsets are complete and the balanced hull of a compact subset is again compact. Also, a Banach space can be complete without being weakly complete.
If is bounded then is absorbing in . If is a locally convex space and is absorbing in then is bounded in Moreover, a subset is weakly bounded if and only if is absorbing in For this reason, it is common to restrict attention to families of bounded subsets of

Weak/weak* topology

The topology has the following properties:

Banach–Alaoglu theorem: Every equicontinuous subset of is relatively compact for
- it follows that the -closure of the convex balanced hull of an equicontinuous subset of is equicontinuous and -compact.
Theorem : Suppose that and are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that is a continuous linear map. Then is surjective if and only if the transpose of is one-to-one and the image of is weakly closed in
Suppose that and are Fréchet spaces, is a Hausdorff locally convex space and that is a separately-continuous bilinear map. Then is continuous.
- In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
is normable if and only if is finite-dimensional.
When is infinite-dimensional the topology on is strictly coarser than the strong dual topology
Suppose that is a locally convex Hausdorff space and that is its completion. If then is strictly finer than
Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the topology.
If is locally convex then a subset is -bounded if and only if there exists a barrel in such that

Compact-convex convergence

If is a Fréchet space then the topologies

Compact convergence

If is a Fréchet space or a LF-space then is complete.
Suppose that is a metrizable topological vector space and that If the intersection of with every equicontinuous subset of is weakly-open, then is open in

Precompact convergence

Banach–Alaoglu theorem: An equicontinuous subset has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on coincides with the topology.

Mackey topology {{math|τ(''X'', ''X'')}}

By letting be the set of all convex balanced weakly compact subsets of will have the Mackey topology on or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by and with this topology is denoted by

Strong dual topology

Due to the importance of this topology, the continuous dual space of is commonly denoted simply by Consequently,
The topology has the following properties:

If is locally convex, then this topology is finer than all other -topologies on when considering only 's whose sets are subsets of
If is a bornological space then is complete.
If is a normed space then the strong dual topology on may be defined by the norm where
If is a LF-space that is the inductive limit of the sequence of space then is a Fréchet space if and only if all are normable.
If is a Montel space then
- has the Heine–Borel property
- On bounded subsets of the strong and weak topologies coincide.
- Every weakly convergent sequence in is strongly convergent.

Mackey topology {{math|τ(''X'', ''X'')}}

By letting be the set of all convex balanced weakly compact subsets of will have the Mackey topology on induced by or the topology of uniform convergence on convex balanced weakly compact subsets of , which is denoted by and with this topology is denoted by

This topology is finer than and hence finer than

Polar topologies induced by subsets of the continuous dual space

Throughout, will be a TVS over the field with continuous dual space and the canonical pairing will be associated with and The table below defines many of the most common polar topologies on

	Notation	Name	Alternative name
finite subsets of		pointwise/simple convergence	weak topology
equicontinuous subsets		equicontinuous convergence
weak-* compact disks			Mackey topology
weak-* compact convex subsets		compact convex convergence
weak-* compact subsets		compact convergence
weak-* bounded subsets		bounded convergence	strong topology

The closure of an equicontinuous subset of is weak-* compact and equicontinuous and furthermore, the convex balanced hull of an equicontinuous subset is equicontinuous.

Weak topology {{math|?(''X'', ''X'')}}

Suppose that and are Hausdorff locally convex spaces with metrizable and that is a linear map. Then is continuous if and only if is continuous. That is, is continuous when and carry their given topologies if and only if is continuous when and carry their weak topologies.

Convergence on equicontinuous sets {{math|?(''X'', ''X'')}}

If was the set of all convex balanced weakly compact equicontinuous subsets of then the same topology would have been induced.
If is locally convex and Hausdorff then 's given topology is exactly
That is, for Hausdorff and locally convex, if then is equicontinuous if and only if is equicontinuous and furthermore, for any is a neighborhood of the origin if and only if is equicontinuous.
Importantly, a set of continuous linear functionals on a TVS is equicontinuous if and only if it is contained in the polar of some neighborhood of the origin in . Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of "encode" all information about 's topology. Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of ".

Mackey topology {{math|τ(''X'', ''X'')}}

Suppose that is a locally convex Hausdorff space. If is metrizable or barrelled then 's original topology is identical to the Mackey topology

Topologies compatible with pairings

Let be a vector space and let be a vector subspace of the algebraic dual of that separates points on If is any other locally convex Hausdorff topological vector space topology on then is compatible with duality between and if when is equipped with then it has as its continuous dual space. If is given the weak topology then is a Hausdorff locally convex topological vector space and is compatible with duality between and .
The question arises: what are all of the locally convex Hausdorff TVS topologies that can be placed on that are compatible with duality between and ? The answer to this question is called the Mackey–Arens theorem.