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 (either the real numbers or complex numbers) 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
Notation: For all let denote the linear functional on defined by and let Similarly, for all let be defined by and let
The weak topology on induced by (and ) 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 (and ), 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
Convention and Definition: Given any definition for a pairing one obtains a dual definition by applying it to the pairing If the definition depends on the order of and (e.g. the definition of "the weak topology defined on by ") then by switching the order of and it is meant that this definition should be applied to (e.g. this gives us the definition of "the weak topology defined on by ").
For instance, after defining " distinguishes points of " (resp, " is a total subset of ") as above, then the dual definition of " distinguishes points of " (resp, " is a total subset 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.
Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) is given for a pairing then mention the corresponding dual definition (or result) will be omitted but it may nevertheless be used.
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:
Convention: This article will use the common practice of treating a pairing interchangeably with and also denoting by
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 (or generated) by (and ), also called the -topology on or the topology of uniform convergence on the sets of is the unique topological vector space (TVS) 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 (i.e. if for all there exists some such that ) 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
The results in the article Topologies on spaces of linear maps can be applied to polar topologies.
Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of
Hausdorffness
We say that covers if every point in belong to some set in
We say that is total in if the linear span of is dense in
Proof of (2): 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 (non-zero) such that where (since is surjective) it can be assumed without loss of generality that The set is a -open subset of that is not empty (since it contains ). 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 (i.e. the topology generated by -complete and bounded disks) 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: If denotes a polar topology on then endowed with this topology will be denoted by or simply For example, if then so that and all denote with endowed with
("topology of uniform convergence on ...") Notation Name ("topology of...") Alternative name finite subsets of (or -closed disked hulls of finite subsets of ) pointwise/simple convergence weak/weak* topology -compact disks Mackey topology -compact convex subsets compact convex convergence -compact subsets(or balanced -compact subsets) compact convergence -complete and bounded disks convex balanced complete bounded convergence -precompact/totally bounded subsets(or balanced -precompact subsets) precompact convergence -infracomplete and bounded disks convex balanced infracomplete bounded convergence -bounded subsets bounded convergence strong topologyStrongest 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 (which implies that is complex valued) 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
for all
Mackey topology τ(Y, X)
The continuous dual space of is (in the exact same way as was described for the weak topology). 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 𝛽(Y, X)
A neighborhood basis (not just a subbasis) 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 (TVS) 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 (this necessarily happens if, for instance, is not Hausdorff), 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 (respectively, ) topology and carry any topology stronger than the topology (respectively, ). 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: If denotes a polar topology then endowed with this topology will be denoted by (e.g. if then and so that denotes with endowed with ). If in addition, then this TVS may be denoted by (for example, ).