This article is about dual pairs of vector spaces. For dual pairs in representation theory, see Reductive dual pair.
In the field of functional analysis, a subfield of mathematics, a dual system, dual pair, or a duality over a field ( is either the real or the complex numbers) is a triple consisting of two vector spaces over and a bilinear map such that for all non-zero the map is not identically 0 and for all non-zero the map is not identically 0.
The study of dual systems is called duality theory.
According to Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the deepest and most beautiful results of the subject."[1]
Definition, notation, and conventions
Pairings
A pairing or a pair over a field is a triple which may also be denoted by consisting of two vector spaces and over (which this article assumes is either the real numbers or the complex numbers) and is a bilinear map which is called the bilinear map associated with the pairing[2] or simply the pairing's map/bilinear form.
For every define
and for every define
Every is a linear functional on and every is a linear functional on
Let
where each of these sets forms a vector space of linear functionals.
It is common practice to write instead of in which case the pair is often denoted by rather than However, this article will reserve use of for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.
Dual pairing/system
A pairing is called a dual system, a dual pair,[3] or a duality over if the bilinear form is non-degenerate, which means that it satisfies the following two separation axioms:
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 ;
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
In this case say that is non-degenerate, say that places and in duality (or in separated duality), and is called the duality pairing of the [2][3]
Total subsets
A subset of is called total if for every implies
A total subset of is defined analogously (see footnote).[note 1]
Orthogonality
The vectors and are called 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.
Throughout, will be a pairing over
The absolute polar or polar of a subset of is the set:[4]
Dually, the absolute polar or polar of a subset of is denoted by and defined by
In this case, the absolute polar of a subset of is also called the absolute prepolar or prepolar of and may be denoted by
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 [5]
If then the bipolar of denoted by is the set
Similarly, if then the bipolar of is
If is a vector subspace of then and this is also equal to the real polar of
Dual definitions and results
Given a pairing define a new pairing where for all [2]
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 This conventions also apply to theorems.
Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) for a pairing is given then this article will omit mention of the corresponding dual definition (or result) but nevertheless use it.
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 it allows us to avoid having to assign a symbol to
Convention and Notation: If a definition and its notation for a pairing depends on the order of and (e.g. the definition of the Mackey topology on ) then by switching the order of and then it is meant that definition applied to (e.g. actually denotes the topology ).
For instance, once the weak topology on is defined, which is denoted by then this definition will automatically be applied to the pairing so as to obtain the definition of the weak topology on where this topology will be denoted by rather than
Identification of with
Although it is technically incorrect and an abuse of notation, this article will also adhere to the following nearly ubiquitous convention of treating a pairing interchangeably with and also of denoting by
Examples
Restriction of a pairing
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 restrictions 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
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 ).[2]
Canonical duality on a topological vector space
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.[3]
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
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[2] — 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
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 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.[2]
Suppose that is a complex pre-Hilbert space with multiplication denoted 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 rather than 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
Suppose and
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.[3]
A sequence space and its beta dual with the bilinear map defined as for forms a dual system.
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 [2] 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 will arise. 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 [2]
If and is a net in then -converges to if converges to in [2]
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).[2]
If is a pairing and is a proper vector subspace of such that is a dual pair, then is strictly coarser than [2]
Bounded subsets
A subset of is -bounded if and only if
where
Hausdorffness
If is a pairing then the following are equivalent:
distinguishes points of ;
The map defines an injection from into the algebraic dual space of ;[2]
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).
If is a normed space then under the canonical duality, is norm closed in and is norm closed in [2]
Subspaces
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 [6]
Furthermore, if is a dual system then so is [6]
Quotients
Suppose that is a vector subspace of
Then is a paired space where is defined by
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
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."[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 [7]
Transposes
Transpose of a linear map with respect to pairings
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 of adjoint of with respect to and (this should not to 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 for details).
Properties of the transpose
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 [2]
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 [2]
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 [2]
Weak continuity
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:
is weakly continuous (that is, is continuous);
;
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
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.[2]
Weak completeness
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).[2]
If is a vector space then under the canonical duality, is complete.[2]
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.[2]
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 Hausdorfflocally 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
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.[3]
Algebraic adjoint
In the spacial 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 [2][8] where the defining condition for is:
or equivalently,
Examples
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
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:[2]
is continuous;
the transpose of F, with respect to and is well-defined.
If is weakly continuous then will be continuous and furthermore, [8]
A map between topological spaces is relatively open if is an open mapping, where is the range of [2]
Suppose that and are dual systems and is a weakly continuous linear map.
Then the following are equivalent:[2]
is relatively open;
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
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:[2]
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
Let be a locally convex space with continuous dual space and let [2]
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
If is separable and is equicontinuous then when endowed with the subspace topology induced by is metrizable.
If is separable and metrizable, then is separable.
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
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
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
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.[2]
When is endowed with this -topology then it is denoted by Y.
Every polar topology is necessarily locally convex.[2]
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.[2]
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 )
A linear map is Mackey continuous (with respect to and ) if is continuous.[2]
A linear map is strongly continuous (with respect to and ) if is continuous.[2]
Bounded subsets
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
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: [2]
Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff,[3][9] 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 [2]
The following is one of the most important theorems in duality theory.
Mackey-Arens theorem I[2] — 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[2] — 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
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.[10]
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 [2]
In particular, if is a subset of then is a barrel in if and only if it is a barrel in [2]
The following theorem shows that barrels (i.e. closed absorbingdisks) are exactly the polars of weakly bounded subsets.
Theorem[2] — 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
A closed absorbing and balanced subset of absorbs each convex compact subset of (i.e. there exists a real such that contains that set).
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[11][2] — 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
Examples
Space of finite sequences
Let denote the space of all sequences of scalars such that for all sufficiently large
Let and define a bilinear map by
Then [2]
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 ).[2]
It follows that there are weakly bounded (that is, -bounded) subsets of that are not strongly bounded (that is, not -bounded).
Strong dual space – Continuous dual space endowed with the topology of uniform convergence on bounded sets
Strong topology (polar topology) – Continuous dual space endowed with the topology of uniform convergence on bounded setsPages displaying short descriptions of redirect targets
^That is linear in its first coordinate is obvious. Suppose is a scalar. Then which shows that is linear in its second coordinate.
^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 ).
^If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is:
^If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is:
^If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is:
^If is a linear map then 's transpose, is well-defined if and only if distinguishes points of and In this case, for each the defining condition for is:
^Of course, there is an analogous definition for topologies on to be "compatible it a pairing" but this article will only deal with topologies on
^Recall that a collection of subsets of a set is said to cover if every point of is contained in some set belonging to the collection.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. OCLC144216834.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN0-12-585050-6.