Dual system

Summary

In mathematics, a dual system, dual pair or a duality over a field is a triple consisting of two vector spaces, and , over and a non-degenerate bilinear map .

Mathematical duality is the study of dual systems and is important in functional analysis. It has extensive applications to quantum mechanics arising in the theory of Hilbert spaces. In addition, it can be very helpful when working with quantum mechanics, quantum physics and much more.

Definition, notation, and conventions

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Pairings

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A pairing or pair over a field is a triple which may also be denoted by consisting of two vector spaces and over and a bilinear map called the bilinear map associated with the pairing,[1] or more simply the pairing's map or its bilinear form. The examples here only describe when is either the real numbers or the complex numbers .

For every , define and for every define Every is a linear functional on and every is a linear functional on . Therefore both form vector spaces of linear functionals.

It is common practice to write instead of , in which in some cases the pairing may be denoted by rather than . However, this article will reserve the use of for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.

Dual pairings

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A pairing is called a dual system, a dual pair,[2] or a duality over if the bilinear form is non-degenerate, which means that 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 (i.e. there exists a such that for each );
  2. separates (distinguishes) points of : if is such that then ; or equivalently, for all non-zero the map is not identically (i.e. there exists an such that for each ).

In this case is non-degenerate, and one can say that places and in duality (or, redundantly but explicitly, in separated duality), and is called the duality pairing of the triple .[1][2]

Total subsets

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A subset of is called total if for every , implies A total subset of is defined analogously (see footnote).[note 1] Thus separates points of if and only if is a total subset of , and similarly for .

Orthogonality

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The vectors and are orthogonal, written , if . Two subsets and are orthogonal, written , if ; that is, if for all and . The definition of a subset being orthogonal to a vector is defined analogously.

The orthogonal complement or annihilator of a subset is Thus is a total subset of if and only if equals .

Polar sets

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Given a triple defining a pairing over , the absolute polar set or polar set of a subset of is the set:Symmetrically, the absolute polar set or polar set of a subset of is denoted by and defined by


To use bookkeeping that helps keep track of the anti-symmetry of the two sides of the duality, the absolute polar of a subset of may also be called the absolute prepolar or prepolar of and then may be denoted by [3]

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 [4]

If is a vector subspace of then and this is also equal to the real polar of If then the bipolar of , denoted , is the polar of the orthogonal complement of , i.e., the set Similarly, if then the bipolar of is

Dual definitions and results

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Given a pairing define a new pairing where for all and .[1]

There is a consistent theme in duality theory 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 These conventions also apply to theorems.

For instance, if " distinguishes points of " (resp, " is a total subset of ") is defined as above, then this convention immediately produces the dual definition of " distinguishes points of " (resp, " is a total subset of ").

This following notation is almost ubiquitous and allows us to avoid assigning a symbol to

Convention and Notation: If a definition and its notation for a pairing depends on the order of and (for example, the definition of the Mackey topology on ) then by switching the order of and then it is meant that definition applied to (continuing the same example, the topology would actually denote the topology ).

For another example, once the weak topology on is defined, denoted by , then this dual definition would automatically be applied to the pairing so as to obtain the definition of the weak topology on , and this topology would be denoted by rather than .

Identification of with

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Although it is technically incorrect and an abuse of notation, this article will adhere to the nearly ubiquitous convention of treating a pairing interchangeably with and also of denoting by

Examples

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Restriction of a pairing

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Suppose that is a pairing, is a vector subspace of and is a vector subspace of . Then the restriction of to is the pairing If is a duality, then it's possible for a restriction to fail to be a duality (e.g. if and ).

This article will use the common practice of denoting the restriction by

Canonical duality on a vector space

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Suppose that is a vector space and let denote the algebraic dual space of (that is, the space of all linear functionals on ). 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 it is instead called 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.

The evaluation map will be denoted by (rather than by ) and will be written rather than

Assumption: As is common practice, if is a vector space and is a vector space of linear functionals on then unless stated otherwise, it will be assumed that they are associated with the canonical pairing

If is a vector subspace of then distinguishes points of (or equivalently, is a duality) if and only if distinguishes points of or equivalently if is total (that is, for all implies ).[1]

Canonical duality on a topological vector space

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Suppose is a topological vector space (TVS) with continuous dual space Then the restriction of the canonical duality to × defines a pairing for which separates points of If separates points of (which is true if, for instance, is a Hausdorff locally convex space) then this pairing forms a duality.[2]

Assumption: As is commonly done, whenever is a TVS, then unless indicated otherwise, it will be assumed without comment that it's associated with the canonical pairing

Polars and duals of TVSs

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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.

Theorem[1] — Let be a TVS with algebraic dual and let be a basis of neighborhoods of at the origin. Under the canonical duality the continuous dual space of is the union of all as ranges over (where the polars are taken in ).

Inner product spaces and complex conjugate spaces

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A pre-Hilbert space is a dual pairing if and only if is vector space over or has dimension Here it is assumed that the sesquilinear form is conjugate homogeneous in its second coordinate and homogeneous in its first coordinate.

  • If is a real Hilbert space then forms a dual system.
  • If is a complex Hilbert space then forms a dual system if and only if If is non-trivial then does not even form pairing since the inner product is sesquilinear rather than bilinear.[1]

Suppose that is a complex pre-Hilbert space with scalar multiplication denoted as usual by juxtaposition or by a dot Define the map where the right-hand side uses the scalar multiplication of Let denote the complex conjugate vector space of where denotes the additive group of (so vector addition in is identical to vector addition in ) but with scalar multiplication in being the map (instead of the scalar multiplication that is endowed with).

The map defined by is linear in both coordinates[note 2] and so forms a dual pairing.

Other examples

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  • Suppose and for all let Then is a pairing such that distinguishes points of but does not distinguish points of Furthermore,
  • Let (where is such that ), and Then is a dual system.
  • Let and be vector spaces over the same field Then the bilinear form places and in duality.[2]
  • A sequence space and its beta dual with the bilinear map defined as for forms a dual system.

Weak topology

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Suppose that is a pairing of vector spaces over If then the weak topology on induced by (and ) is the weakest TVS topology on denoted by or simply making all maps continuous as ranges over [1] If is not clear from context then it should be assumed to be all of in which case it is called the weak topology on (induced by ). The notation or (if no confusion could arise) simply is used to denote endowed with the weak topology Importantly, the weak topology depends entirely on the function the usual topology on and 's vector space structure but not on the algebraic structures of

Similarly, if then the dual definition of the weak topology on induced by (and ), which is denoted by or simply (see footnote for details).[note 3]

Definition and Notation: If "" is attached to a topological definition (e.g. -converges, -bounded, etc.) then it means that definition when the first space (i.e. ) carries the topology. Mention of or even and may be omitted if no confusion arises. So, for instance, if a sequence in "-converges" or "weakly converges" then this means that it converges in whereas if it were a sequence in , then this would mean that it converges in ).

The topology is locally convex since it is determined by the family of seminorms defined by as ranges over [1] If and is a net in then -converges to if converges to in [1] A net -converges to if and only if for all converges to If is a sequence of orthonormal vectors in Hilbert space, then converges weakly to 0 but does not norm-converge to 0 (or any other vector).[1]

If is a pairing and is a proper vector subspace of such that is a dual pair, then is strictly coarser than [1]

Bounded subsets

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A subset of is -bounded if and only if where

Hausdorffness

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If is a pairing then the following are equivalent:

  1. distinguishes points of ;
  2. The map defines an injection from into the algebraic dual space of ;[1]
  3. is Hausdorff.[1]

Weak representation theorem

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

Weak representation theorem[1] — Let be a pairing over the field Then the continuous dual space of is Furthermore,

  1. If is a continuous linear functional on then there exists some such that ; if such a exists then it is unique if and only if distinguishes points of
    • Note that whether or not distinguishes points of is not dependent on the particular choice of
  2. The continuous dual space of may be identified with the quotient space where
    • This is true regardless of whether or not distinguishes points of or distinguishes points of

Consequently, the continuous dual space of is

With respect to the canonical pairing, if is a TVS whose continuous dual space separates points on (i.e. such that is Hausdorff, which implies that is also necessarily Hausdorff) then the continuous dual space of is equal to the set of all "evaluation at a point " maps as ranges over (i.e. the map that send to ). This is commonly written as This very important fact is why results for polar topologies on continuous dual spaces, such as the strong dual topology on for example, can also often be applied to the original TVS ; for instance, being identified with means that the topology on can instead be thought of as a topology on Moreover, if is endowed with a topology that is finer than then the continuous dual space of will necessarily contain as a subset. So for instance, when is endowed with the strong dual topology (and so is denoted by ) then which (among other things) allows for to be endowed with the subspace topology induced on it by, say, the strong dual topology (this topology is also called the strong bidual topology and it appears in the theory of reflexive spaces: the Hausdorff locally convex TVS is said to be semi-reflexive if and it will be called reflexive if in addition the strong bidual topology on is equal to 's original/starting topology).

Orthogonals, quotients, and subspaces

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If is a pairing then for any subset of :

  • and this set is -closed;[1]
  • ;[1]
    • Thus if is a -closed vector subspace of then
  • If is a family of -closed vector subspaces of then [1]
  • If is a family of subsets of then [1]

If is a normed space then under the canonical duality, is norm closed in and is norm closed in [1]

Subspaces

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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 means ) where is defined by

The topology is equal to the subspace topology that inherits from [5] Furthermore, if is a dual system then so is [5]

Quotients

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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 [5]

Polars and the weak topology

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If is a locally convex space and if is a subset of the continuous dual space then is -bounded if and only if for some barrel in [1]

The following results are important for defining polar topologies.

If is a pairing and then:[1]

  1. The polar of is a closed subset of
  2. The polars of the following sets are identical: (a) ; (b) the convex hull of ; (c) the balanced hull of ; (d) the -closure of ; (e) the -closure of the convex balanced hull of
  3. The bipolar theorem: The bipolar of denoted by is equal to the -closure of the convex balanced hull of
    • The bipolar theorem in particular "is an indispensable tool in working with dualities."[4]
  4. is -bounded if and only if is absorbing in
  5. If in addition distinguishes points of then is -bounded if and only if it is -totally bounded.

If is a pairing and is a locally convex topology on that is consistent with duality, then a subset of is a barrel in if and only if is the polar of some -bounded subset of [6]

Transposes

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Transposes of a linear map with respect to pairings

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Let and be pairings over and let be a linear map.

For all let be the map defined by It is said that 's transpose or adjoint is well-defined if the following conditions are satisfied:

  1. distinguishes points of (or equivalently, the map from into the algebraic dual is injective), and
  2. where and .

In this case, for any there exists (by condition 2) a unique (by condition 1) such that ), where this element of will be denoted by This defines a linear map

called the transpose or adjoint of with respect to and (this should not be confused with the Hermitian adjoint). It is easy to see that the two conditions mentioned above (i.e. for "the transpose is well-defined") are also necessary for to be well-defined. For every the defining condition for is that is,      for all

By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form [note 4] [note 5] [note 6] [note 7] etc. (see footnote).

Properties of the transpose

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Throughout, and be pairings over and will be a linear map whose transpose is well-defined.

  • is injective (i.e. ) if and only if the range of is dense in [1]
  • If in addition to being well-defined, the transpose of is also well-defined then
  • Suppose is a pairing over and is a linear map whose transpose is well-defined. Then the transpose of which is is well-defined and
  • If is a vector space isomorphism then is bijective, the transpose of which is is well-defined, and [1]
  • Let and let denotes the absolute polar of then:[1]
    1. ;
    2. if for some then ;
    3. if is such that then ;
    4. if and are weakly closed disks then if and only if ;
These results hold when the real polar is used in place of the absolute polar.

If and are normed spaces under their canonical dualities and if is a continuous linear map, then [1]

Weak continuity

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A linear map is weakly continuous (with respect to and ) if is continuous.

The following result shows that the existence of the transpose map is intimately tied to the weak topology.

Proposition — Assume that distinguishes points of and is a linear map. Then the following are equivalent:

  1. is weakly continuous (that is, is continuous);
  2. ;
  3. the transpose of is well-defined.

If is weakly continuous then

  • is weakly continuous, meaning that is continuous;
  • the transpose of is well-defined if and only if distinguishes points of in which case

Weak topology and the canonical duality

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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.[1]

Weak completeness

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If is a complete topological vector space say that is -complete or (if no ambiguity can arise) weakly-complete. There exist Banach spaces that are not weakly-complete (despite being complete in their norm topology).[1]

If is a vector space then under the canonical duality, is complete.[1] Conversely, if is a Hausdorff locally convex TVS with continuous dual space then is complete if and only if ; that is, if and only if the map defined by sending to the evaluation map at (i.e. ) is a bijection.[1]

In particular, with respect to the canonical duality, if is a vector subspace of such that separates points of then is complete if and only if Said differently, there does not exist a proper vector subspace of such that is Hausdorff and is complete in the weak-* topology (i.e. the topology of pointwise convergence). Consequently, when the continuous dual space of a Hausdorff locally convex TVS is endowed with the weak-* topology, then is complete if and only if (that is, if and only if every linear functional on is continuous).

Identification of Y with a subspace of the algebraic dual

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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 (where is the natural evaluation map). In particular, in this situation it will be assumed without loss of generality that is a vector subspace of 's algebraic dual and is the evaluation map.

Convention: Often, whenever is injective (especially when forms a dual pair) then it is common practice to assume without loss of generality that is a vector subspace of the algebraic dual space of that is the natural evaluation map, and also denote by

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.[2]

Algebraic adjoint

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In the special 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 and it will be denoted by ; that is, In this case, for all [1][7] where the defining condition for is: or equivalently,

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 of

Weak continuity and openness

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Suppose that and are canonical pairings (so and ) that are dual systems and let be a linear map. Then is weakly continuous if and only if it satisfies any of the following equivalent conditions:[1]

  1. is continuous.
  2. the transpose of F, with respect to and is well-defined.

If is weakly continuous then will be continuous and furthermore, [7]

A map between topological spaces is relatively open if is an open mapping, where is the range of [1]

Suppose that and are dual systems and is a weakly continuous linear map. Then the following are equivalent:[1]

  1. is relatively open.
  2. The range of is -closed in ;

Furthermore,

  • is injective (resp. bijective) if and only if is surjective (resp. bijective);
  • is surjective if and only if is relatively open and injective.
Transpose of a map between TVSs
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The transpose of map between two TVSs is defined if and only if is weakly continuous.

If is a linear map between two Hausdorff locally convex topological vector spaces, then:[1]

  • If is continuous then it is weakly continuous and is both Mackey continuous and strongly continuous.
  • If is weakly continuous then it is both Mackey continuous and strongly continuous (defined below).
  • If is weakly continuous then it is continuous if and only if maps equicontinuous subsets of to equicontinuous subsets of
  • If and are normed spaces then is continuous if and only if it is weakly continuous, in which case
  • If is continuous then is relatively open if and only if is weakly relatively open (i.e. is relatively open) and every equicontinuous subsets of is the image of some equicontinuous subsets of
  • If is continuous injection then is a TVS-embedding (or equivalently, a topological embedding) if and only if every equicontinuous subsets of is the image of some equicontinuous subsets of

Metrizability and separability

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Let be a locally convex space with continuous dual space and let [1]

  1. If is equicontinuous or -compact, and if is such that is dense in then the subspace topology that inherits from is identical to the subspace topology that inherits from
  2. If is separable and is equicontinuous then when endowed with the subspace topology induced by is metrizable.
  3. If is separable and metrizable, then is separable.
  4. If is a normed space then is separable if and only if the closed unit call the continuous dual space of is metrizable when given the subspace topology induced by
  5. If is a normed space whose continuous dual space is separable (when given the usual norm topology), then is separable.

Polar topologies and topologies compatible with pairing

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Starting with only the weak topology, the use of polar sets produces a range of locally convex topologies. 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

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Given a collection of subsets of , the polar topology on determined by (and ) or the -topology on is the unique topological vector space (TVS) topology on for which forms a subbasis of neighborhoods at the origin.[1] When is endowed with this -topology then it is denoted by Y. Every polar topology is necessarily locally convex.[1] When is a directed set with respect to subset inclusion (i.e. if for all there exists some such that ) then this neighborhood subbasis at 0 actually forms a neighborhood basis at 0.[1]

The following table lists some of the more important polar topologies.

Notation: If denotes a polar topology on then endowed with this topology will be denoted by or simply (e.g. for we'd have so that and all denote 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
-bounded subsets
bounded convergence strong topology
Strongest polar topology

Definitions involving polar topologies

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Continuity

A linear map is Mackey continuous (with respect to and ) if is continuous.[1]

A linear map is strongly continuous (with respect to and ) if is continuous.[1]

Bounded subsets

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A subset of is weakly bounded (resp. Mackey bounded, strongly bounded) if it is bounded in (resp. bounded in bounded in ).

Topologies compatible with a pair

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If is a pairing over and is a vector topology on then is a topology of the pairing and that it is compatible (or consistent) with the pairing if it is locally convex and if the continuous dual space of [note 8] If distinguishes points of then by identifying as a vector subspace of 's algebraic dual, the defining condition becomes: [1] Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff,[2][8] which it would have to be if distinguishes the points of (which these authors assume).

The weak topology is compatible with the pairing (as was shown in the Weak representation theorem) and it is in fact the weakest such topology. There is a strongest topology compatible with this pairing and that is the Mackey topology. 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 [1]

Mackey–Arens theorem

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The following is one of the most important theorems in duality theory.

Mackey–Arens theorem I[1] — Let will be a pairing such that distinguishes the points of and let be a locally convex topology on (not necessarily Hausdorff). Then is compatible with the pairing if and only if is a polar topology determined by some collection of -compact disks that cover[note 9]

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–Arens theorem II[1] — Let will be a pairing such that distinguishes the points of and let be a locally convex topology on Then is compatible with the pairing if and only if

Mackey's theorem, barrels, and closed convex sets

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If is a TVS (over or ) then a half-space is a set of the form for some real and some continuous real linear functional on

Theorem — If is a locally convex space (over or ) and if is a non-empty closed and convex subset of then is equal to the intersection of all closed half spaces containing it.[9]

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 [1] In particular, if is a subset of then is a barrel in if and only if it is a barrel in [1]

The following theorem shows that barrels (i.e. closed absorbing disks) are exactly the polars of weakly bounded subsets.

Theorem[1] — Let will be a pairing such that distinguishes the points of and let be a topology of the pair. Then a subset of is a barrel in if and only if it is equal to the polar of some -bounded subset of

If is a topological vector space, then:[1][10]

  1. A closed absorbing and balanced subset of absorbs each convex compact subset of (i.e. there exists a real such that contains that set).
  2. If is Hausdorff and locally convex then every barrel in absorbs every convex bounded complete subset of

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.

Mackey's theorem[10][1] — Suppose that is a Hausdorff locally convex space with continuous dual space and consider the canonical duality If is any topology on that is compatible with the duality on then the bounded subsets of are the same as the bounded subsets of

Space of finite sequences

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Let denote the space of all sequences of scalars such that for all sufficiently large Let and define a bilinear map by Then [1] Moreover, a subset is -bounded (resp. -bounded) if and only if there exists a sequence of positive real numbers such that for all and all indices (resp. and ).[1]

It follows that there are weakly bounded (that is, -bounded) subsets of that are not strongly bounded (that is, not -bounded).

See also

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Notes

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  1. ^ A subset of is total if for all , implies .
  2. ^ That is linear in its first coordinate is obvious. Suppose is a scalar. Then which shows that is linear in its second coordinate.
  3. ^ The weak topology on is the weakest TVS topology on making all maps continuous, as ranges over The dual notation of or simply may also be used to denote endowed with the weak topology If is not clear from context then it should be assumed to be all of in which case it is simply called the weak topology on (induced by ).
  4. ^ If is a linear map then 's transpose,