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 .
In mathematics, duality is the study of dual systems and is important in functional analysis. Duality plays crucial roles in quantum mechanics because it has extensive applications to the theory of Hilbert spaces.
Definition, notation, and conventions
Pairings A 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 more simply called the pairing's map or its bilinear form. The examples here only describe when is either the real numbers or the complex numbers , but the mathematical theory is general.
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 A pairing is called a , a , or a 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 A subset of is called if for every , implies A total subset of is defined analogously (see footnote).A subset of is total if for all , implies . Thus separates points of if and only if is a total subset of , and similarly for .
Orthogonality 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
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 .
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 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
: Given any definition for a pairing one obtains a 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
: 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 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
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 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 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 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 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.
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 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 coordinatesThat is linear in its first coordinate is obvious. Suppose is a scalar. Then which shows that is linear in its second coordinate. and so forms a dual pairing.
Other examples 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
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 each map continuous as a function of for every . 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 on the function the usual topology on and 's vector space structure but 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).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 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 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 ; is Hausdorff.
Weak representation theorem The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space 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