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.
Suppose that is a pairing of vector spaces over and for all, let be defined by and for all, let be defined by x ↦ b.
The weak topology on induced by Y is the weakest TVS topology on, denoted by or simply, making all maps continuous, as y ranges over.
Similarly, we have 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.
;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 :=.
Similarly, if then the bipolar of is.
;Boundedness
It is because of the following theorem that we will almost always assume that consists of -bounded subsets of.
Theorem: For any subset, the following are equivalent:
- -bounded ;
- for all, < ∞, where we may also denote this supremum by ;
- is an absorbing subset of.
Dual definitions and results
Given a pairing we can define a new pairing where := b.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 we immediately obtain the dual definition of " distinguishes points of ".
Once we define, for instance, then we will automatically assume that has been defined without mentioning the analogous definition.
The same applies to many theorems.
In particular, although we will only define the general notion of polar topologies on with being a collection of -bounded subsets of, we 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, we will also adhere to the following nearly ubiquitous convention:
Polar topologies
Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of.Observe that for every and, is convex and balanced and since G is -bounded, 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 0 then one may replace the defining neighborhood subbasis at 0 with
without changing the resulting topology.
;Seminorms defining the polar topology
Every determines a seminorm defined by
where observe that and that is in fact the gauge of.
Thus the -topology on is always locally convex.
;Modifying
If every positive scalar multiple of a set in is contained in some set belonging to then the defining neighborhood subbasis at 0 can be replaced with
without changing the resulting topology.
When is a directed set with respect to subset inclusion then the defining neighborhood subbasis at 0 actually forms a neighborhood basis at 0.
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 to satisfy the following conditions:
- the union of any two sets is contained in some set ;
- all scalar multiples of every belongs to.
;Convergence of nets and filters
If is a net in then in the -topology on if and only if for all,
pG → 0, or in words, if and only if for all, the net of linear functionals on converges uniformly to 0 on G.
If then in the -topology on if and only if for all,
pG → 0.
A filter on converges to an element in the -topology on if converges uniformly to y on each.
Properties
Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of.;Hausdorffness
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.In the following table, we 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 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, for all.
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, we will let be a topological vector space with continuous dual space and will be the canonical pairing.Note that 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 is continuous on.
- If in addition separates points on then the canonical map of into is an injection.
- In particular, the transpose of is continuous if carries the topology and carry any topology stronger than the topology.
Polar topologies on the continuous dual space
Throughout, will be a TVS over the field with continuous dual space and we will associate and 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 | |
| σ-compact disks in | Mackey topology |
To see why some of the above collections induce the same polar topologies, we recall that facts.
Recall that 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.
Therefore, we restrict our attention to families of bounded subsets of.
Weak/weak* topology σ(''X''{{big|}}, ''X'')
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.
- In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
Compact-convex convergence {{math|γ(''X''{{big|}}, ''X'')}}
- If is a Fréchet space then the topologies.
Compact convergence c(''X''{{big|}}, ''X'')
- 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 K of has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on K coincides with the topology.
Mackey topology τ(''X''{{big|}}, ''X'')
Strong dual topology b(''X''{{big|}}, ''X'')
Due to the importance of this topology, the continuous dual space of is commonly denoted by .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 τ(, {{mvar|X{{big|}}}})
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 we will associate and 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 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 |
Recall that 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''{{big|}})}}
- 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 ε(''X'', ''X''{{big|}})
- 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.
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 0 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 τ(''X'', ''X''{{big|}})
- 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
If is any other locally convex Hausdorff topological vector space topology on, then we say that is compatible with duality between and if when is equipped with, then it has as its continuous dual space.
If we give the weak topology then is a Hausdorff locally convex topological vector space and is compatible with duality between and .
We can now ask the question: what are all of the locally convex Hausdorff TVS topologies that we can place on that are compatible with duality between and ?
The answer to this question is called the Mackey–Arens theorem.