This article is about dual pairs of vector spaces. For dual pairs in representation theory, see
Reductive dual pair.
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 theory, the study of dual systems, has an important place in functional analysis and has extensive applications to quantum mechanics via the theory of Hilbert spaces.
Definition, notation, and conventions edit
Pairings edit
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 or simply the pairing's map or its bilinear form. For simplicity, this article only covers examples where 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
. 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 pairing may often 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 edit
A pairing is called a dual system, a dual pair, 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 for each );
- 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 .
Total subsets edit
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 edit
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.
The orthogonal complement or annihilator of a subset is
. Thus
is a total subset of
if and only if
equals
.
Polar sets edit
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 antisymmetry 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
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
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 edit
Given a pairing define a new pairing where for all and .
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 This 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 edit
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 edit
Restriction of a pairing edit
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 edit
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 ).
Canonical duality on a topological vector space edit
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.
- 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 edit
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 — 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 edit
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.
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 edit
- 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.
- A sequence space and its beta dual with the bilinear map defined as for forms a dual system.
Weak topology edit
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 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 If and is a net in then -converges to if converges to in 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).
If is a pairing and is a proper vector subspace of such that is a dual pair, then is strictly coarser than
Bounded subsets edit
A subset of is -bounded if and only if
where
Hausdorffness edit
If is a pairing then the following are equivalent:
- distinguishes points of ;
- The map defines an injection from into the algebraic dual space of ;
- is Hausdorff.
Weak representation theorem edit
The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of
Weak representation theorem — Let be a pairing over the field Then the continuous dual space of is
Furthermore,
- 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
- 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 edit
If is a pairing then for any subset of :
- and this set is -closed;
- ;
- Thus if is a -closed vector subspace of then
- If is a family of -closed vector subspaces of then
- If is a family of subsets of then
If is a normed space then under the canonical duality, is norm closed in and is norm closed in
Subspaces edit
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 Furthermore, if is a dual system then so is
Quotients edit
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
Polars and the weak topology edit
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
The following results are important for defining polar topologies.
If is a pairing and then:
- The polar of is a closed subset of
- 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."
- is -bounded if and only if is absorbing in
- 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
Transposes edit
Transposes of a linear map with respect to pairings edit
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:
- distinguishes points of (or equivalently, the map from into the algebraic dual is injective), and
- where and .
In this case, for any