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Polar topology

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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

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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:

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

Polars

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The polar or absolute polar of a subset is the set[1]

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.[2]

If then the bipolar of denoted by is defined by Similarly, if then the bipolar of is defined to be

Weak topologies

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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 [3] 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 [3]

Weak boundedness and absorbing polars

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It is because of the following theorem that it is almost always assumed that the family consists of -bounded subsets of [3]

Theorem — For any subset the following are equivalent:

  1. is an absorbing subset of
    • If this condition is not satisfied then can not possibly be a neighborhood of the origin in any TVS topology on ;
  2. is a -bounded set; said differently, is a bounded subset of ;
  3. for all where this supremum may also be denoted by

The -bounded subsets of have an analogous characterization.

Dual definitions and results

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Every pairing can be associated with a corresponding pairing where by definition [3]

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

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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.[3] 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 [3]

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.[3]

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

Theorem[3] — Let is a pairing of vector spaces over and let be a non-empty collection of -bounded subsets of The -topology on is not altered if is replaced by any of the following collections of [-bounded] subsets of :

  1. all subsets of all finite unions of sets in ;
  2. all scalar multiples of all sets in ;
  3. the balanced hull of every set in ;
  4. the convex hull of every set in ;
  5. the -closure of every set in ;
  6. the -closure of the convex balanced hull of every set in

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[4] 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

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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 [5] if the linear span of is dense in

Theorem — Let be a pairing of vector spaces over the field and be a non-empty collection of -bounded subsets of Then,

  1. If covers then the -topology on is Hausdorff.[3]
  2. If distinguishes points of and if is a -dense subset of then the -topology on is Hausdorff.[2]
  3. If is a dual system (rather than merely a pairing) then the -topology on is Hausdorff if and only if span of is dense in [3]
Proof

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

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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 topology
Strongest polar topology

Weak topology σ(Y, X)

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For any a basic -neighborhood of in is a set of the form:

for some real and some finite set of points in [3]

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 [3] 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.[3]

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)

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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,[3] 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 [3]

Strong topology 𝛽(Y, X)

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A neighborhood basis (not just a subbasis) at the origin for the topology is:[3]

The strong topology is finer than the Mackey topology.[3]

Polar topologies and topological vector spaces

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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

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  • 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 [6]
    • 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

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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, ).

("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 convex subsets compact convex convergence
compact subsets
(or balanced compact subsets)
compact convergence
-compact disks Mackey topology
precompact/totally bounded subsets
(or balanced precompact 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 (in the same row) 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.[7] Furthermore, in every TVS, compact subsets are complete[7] and the balanced hull of a compact (resp. totally bounded) subset is again compact (resp. totally bounded).[8] Also, a Banach space can be complete without being weakly complete.

If is bounded then is absorbing in (note that being absorbing is a necessary condition for to be a neighborhood of the origin in any TVS topology on ).[2] 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 σ(X', X)

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The topology has the following properties:

  • Banach–Alaoglu theorem: Every equicontinuous subset of is relatively compact for [9]
    • it follows that the -closure of the convex balanced hull of an equicontinuous subset of is equicontinuous and -compact.
  • Theorem (S. Banach): 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 [3]

Compact-convex convergence γ(X', X)

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If is a Fréchet space then the topologies

Compact convergence c(X', X)

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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

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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 τ(X', X)

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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 b(X', X)

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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 (e.g. metrizable or LF-space) then is complete.
  • If is a normed space then the strong dual topology on may be defined by the norm where [10]
  • If is a LF-space that is the inductive limit of the sequence of space (for ) then is a Fréchet space if and only if all are normable.
  • If is a Montel space then
    • has the Heine–Borel property (i.e. every closed and bounded subset of is compact in )
    • On bounded subsets of the strong and weak topologies coincide (and hence so do all other topologies finer than and coarser than ).
    • Every weakly convergent sequence in is strongly convergent.

Mackey topology τ(X, X'')

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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

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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: If denotes a polar topology on then endowed with this topology will be denoted by or (e.g. for we'd have so that and both 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 topology
equicontinuous subsets
(or equicontinuous disks)
(or weak-* compact equicontinuous disks)
equicontinuous convergence
weak-* compact disks Mackey topology
weak-* compact convex subsets compact convex convergence
weak-* compact subsets
(or balanced 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 𝜎(X, X')

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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')

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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 (i.e. the topology that started with) 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 (i.e. ). 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 (i.e. distinct TVS topologies on produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS original topology by taking the polars of sets in the collection). Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of ".

Mackey topology τ(X, X')

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Suppose that is a locally convex Hausdorff space. If is metrizable or barrelled then 's original topology is identical to the Mackey topology [11]

Topologies compatible with pairings

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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 (TVS) and is compatible with duality between and (i.e. ). 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.

See also

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References

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  1. ^ Trèves 2006, p. 195.
  2. ^ a b c Trèves 2006, pp. 195–201.
  3. ^ a b c d e f g h i j k l m n o p q r Narici & Beckenstein 2011, pp. 225–273.
  4. ^ Robertson & Robertson 1964, III.2
  5. ^ Schaefer & Wolff 1999, p. 80.
  6. ^ Trèves 2006, pp. 199–200.
  7. ^ a b Narici & Beckenstein 2011, pp. 47–66.
  8. ^ Narici & Beckenstein 2011, pp. 67–113.
  9. ^ Schaefer & Wolff 1999, p. 85.
  10. ^ Trèves 2006, p. 198.
  11. ^ Trèves 2006, pp. 433.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.