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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a dual system, dual pair or a duality over a field \mathbb is a triple (X, Y, b) consisting of two
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s, X and Y, over \mathbb and a non- degenerate
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear map can also be defined for ...
b : X \times Y \to \mathbb. In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, duality is the study of dual systems and is important in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. Duality plays crucial roles in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
because it has extensive applications to the theory of
Hilbert spaces In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
.


Definition, notation, and conventions


Pairings

A or pair over a field \mathbb is a triple (X, Y, b), which may also be denoted by b(X, Y), consisting of two vector spaces X and Y over \mathbb and a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear map can also be defined for ...
b : X \times Y \to \mathbb called the bilinear map associated with the pairing, or more simply called the pairing's map or its
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
. The examples here only describe when \mathbb is either the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s or the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s \Complex, but the mathematical theory is general. For every x \in X, define \begin b(x, \,\cdot\,) : \,& Y && \to &&\, \mathbb \\ & y && \mapsto &&\, b(x, y) \end and for every y \in Y, define \begin b(\,\cdot\,, y) : \,& X && \to &&\, \mathbb \\ & x && \mapsto &&\, b(x, y). \end Every b(x, \,\cdot\,) is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
on Y and every b(\,\cdot\,, y) is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
on X. Therefore both b(X, \,\cdot\,) := \ \qquad \text \qquad b(\,\cdot\,, Y) := \, form vector spaces of
linear functionals In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mat ...
. It is common practice to write \langle x, y \rangle instead of b(x, y), in which in some cases the pairing may be denoted by \left\langle X, Y \right\rangle rather than (X, Y, \langle \cdot, \cdot \rangle). However, this article will reserve the use of \langle \cdot, \cdot \rangle for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.


Dual pairings

A pairing (X, Y, b) is called a , a , or a over \mathbb if the
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
b is non- degenerate, which means that it satisfies the following two separation axioms: # Y separates (distinguishes) points of X: if x \in X is such that b(x, \,\cdot\,) = 0 then x = 0; or equivalently, for all non-zero x \in X, the map b(x, \,\cdot\,) : Y \to \mathbb is not identically 0 (i.e. there exists a y \in Y such that b(x, y) \neq 0 for each x \in X); # X separates (distinguishes) points of Y: if y \in Y is such that b(\,\cdot\,, y) = 0 then y = 0; or equivalently, for all non-zero y \in Y, the map b(\,\cdot\,, y) : X \to \mathbb is not identically 0 (i.e. there exists an x \in X such that b(x, y) \neq 0 for each y \in Y). In this case b is non-degenerate, and one can say that b places X and Y in duality (or, redundantly but explicitly, in separated duality), and b is called the duality pairing of the triple (X, Y, b).


Total subsets

A subset S of Y is called if for every x \in X, b(x, s) = 0 \quad \text s \in S implies x = 0. A total subset of X is defined analogously (see footnote).A subset S of X is total if for all y \in Y, b(s, y) = 0 \quad \text s \in S implies y = 0. Thus X separates points of Y if and only if X is a total subset of X, and similarly for Y.


Orthogonality

The vectors x and y are
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
, written x \perp y, if b(x, y) = 0. Two subsets R \subseteq X and S \subseteq Y are
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
, written R \perp S, if b(R, S) = \; that is, if b(r, s) = 0 for all r \in R and s \in S. The definition of a subset being orthogonal to a vector is defined analogously. The
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^\perp of all vectors in V that are orthogonal to every vector in W. I ...
or annihilator of a subset R \subseteq X is R^ := \ := \Thus R is a total subset of X if and only if R^\perp equals \.


Polar sets

Given a triple (X, Y, b) defining a pairing over \mathbb, the absolute polar set or polar set of a subset A of X is the set:A^ := \left\. Symmetrically, the absolute polar set or polar set of a subset B of Y is denoted by B^ and defined by B^ := \left\. To use bookkeeping that helps keep track of the anti-symmetry of the two sides of the duality, the absolute polar of a subset B of Y may also be called the absolute prepolar or prepolar of B and then may be denoted by ^B. The polar B^ is necessarily a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
set containing 0 \in Y where if B is balanced then so is B^ and if B is a vector subspace of X then so too is B^ a vector subspace of Y. If A is a vector subspace of X, then A^ = A^ and this is also equal to the real polar of A. If A \subseteq X then the bipolar of A, denoted A^, is the polar of the orthogonal complement of A, i.e., the set ^\left(A^\right). Similarly, if B \subseteq Y then the bipolar of B is B^ := \left(^B\right)^.


Dual definitions and results

Given a pairing (X, Y, b), define a new pairing (Y, X, d) where d(y, x) := b(x, y) for all x \in X and y \in Y. There is a consistent theme in duality theory that any definition for a pairing (X, Y, b) has a corresponding dual definition for the pairing (Y, X, d). :: Given any definition for a pairing (X, Y, b), one obtains a by applying it to the pairing (Y, X, d). These conventions also apply to theorems. For instance, if "X distinguishes points of Y" (resp, "S is a total subset of Y") is defined as above, then this convention immediately produces the dual definition of "Y distinguishes points of X" (resp, "S is a total subset of X"). This following notation is almost ubiquitous and allows us to avoid assigning a symbol to d. :: If a definition and its notation for a pairing (X, Y, b) depends on the order of X and Y (for example, the definition of the
Mackey topology In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not ...
\tau(X, Y, b) on X) then by switching the order of X and Y, then it is meant that definition applied to (Y, X, d) (continuing the same example, the topology \tau(Y, X, b) would actually denote the topology \tau(Y, X, d)). For another example, once the weak topology on X is defined, denoted by \sigma(X, Y, b), then this dual definition would automatically be applied to the pairing (Y, X, d) so as to obtain the definition of the weak topology on Y, and this topology would be denoted by \sigma(Y, X, b) rather than \sigma(Y, X, d).


Identification of (X, Y) with (Y, X)

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


Examples


Restriction of a pairing

Suppose that (X, Y, b) is a pairing, M is a vector subspace of X, and N is a vector subspace of Y. Then the restriction of (X, Y, b) to M \times N is the pairing \left(M, N, b\big\vert_\right). If (X, Y, b) is a duality, then it's possible for a restriction to fail to be a duality (e.g. if Y \neq \ and N = \). This article will use the common practice of denoting the restriction \left(M, N, b\big\vert_\right) by (M, N, b).


Canonical duality on a vector space

Suppose that X is a vector space and let X^ denote the algebraic dual space of X (that is, the space of all linear functionals on X). There is a canonical duality \left(X, X^, c\right) where c\left(x, x^\right) = \left\langle x, x^ \right\rangle = x^(x), which is called the evaluation map or the natural or canonical bilinear functional on X \times X^. Note in particular that for any x^ \in X^, c\left(\,\cdot\,, x^\right) is just another way of denoting x^; i.e. c\left(\,\cdot\,, x^\right) = x^(\,\cdot\,) = x^. If N is a vector subspace of X^, then the restriction of \left(X, X^, c\right) to X \times N is called the canonical pairing where if this pairing is a duality then it is instead called the canonical duality. Clearly, X always distinguishes points of N, so the canonical pairing is a dual system if and only if N separates points of X. The following notation is now nearly ubiquitous in duality theory. The evaluation map will be denoted by \left\langle x, x^ \right\rangle = x^(x) (rather than by c) and \langle X, N \rangle will be written rather than (X, N, c). :Assumption: As is common practice, if X is a vector space and N is a vector space of linear functionals on X, then unless stated otherwise, it will be assumed that they are associated with the canonical pairing \langle X, N \rangle. If N is a vector subspace of X^ then X distinguishes points of N (or equivalently, (X, N, c) is a duality) if and only if N distinguishes points of X, or equivalently if N is total (that is, n(x) = 0 for all n \in N implies x = 0).


Canonical duality on a topological vector space

Suppose X is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) with continuous dual space X^. Then the restriction of the canonical duality \left(X, X^, c\right) to X × X^ defines a pairing \left(X, X^, c\big\vert_\right) for which X separates points of X^. If X^ separates points of X (which is true if, for instance, X is a Hausdorff locally convex space) then this pairing forms a duality. :Assumption: As is commonly done, whenever X is a TVS, then unless indicated otherwise, it will be assumed without comment that it's associated with the canonical pairing \left\langle X, X^ \right\rangle.


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 (H, \langle \cdot, \cdot \rangle) is a dual pairing if and only if H is vector space over \R or H has dimension 0. Here it is assumed that the sesquilinear form \langle \cdot, \cdot \rangle is conjugate homogeneous in its second coordinate and homogeneous in its first coordinate. Suppose that (H, \langle \cdot, \cdot \rangle) is a complex pre-Hilbert space with scalar multiplication denoted as usual by juxtaposition or by a dot \cdot. Define the map \,\cdot\, \perp \,\cdot\, : \Complex \times H \to H \quad \text \quad c \perp x := \overline x, where the right-hand side uses the scalar multiplication of H. Let \overline denote the complex conjugate vector space of H, where \overline denotes the additive group of (H, +) (so vector addition in \overline is identical to vector addition in H) but with scalar multiplication in \overline being the map \,\cdot\, \perp \,\cdot\, (instead of the scalar multiplication that H is endowed with). The map b : H \times \overline \to \Complex defined by b(x, y) := \langle x, y \rangle is linear in both coordinatesThat b is linear in its first coordinate is obvious. Suppose c is a scalar. Then b(x, c \perp y) = b\left(x, \overline y\right) = \langle x, \overline y \rangle = c \langle x, y \rangle = c b(x, y), which shows that b is linear in its second coordinate. and so \left(H, \overline, \langle \cdot, \cdot \rangle\right) forms a dual pairing.


Other examples


Weak topology

Suppose that (X, Y, b) is a pairing of
vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and sc ...
over \mathbb. If S \subseteq Y then the weak topology on X induced by S (and b) is the weakest TVS topology on X, denoted by \sigma(X, S, b) or simply \sigma(X, S), making each map b(\,\cdot\,, y) : X \to \mathbb continuous as a function of x for every y \in S . If S is not clear from context then it should be assumed to be all of Y, in which case it is called the weak topology on X (induced by Y). The notation X_, X_, or (if no confusion could arise) simply X_ is used to denote X endowed with the weak topology \sigma(X, S, b). Importantly, the weak topology depends on the function b, the usual topology on \Complex, and X's
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
structure but on the
algebraic structures In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
of Y. Similarly, if R \subseteq X then the dual definition of the weak
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on Y induced by R
(and b), which is denoted by \sigma(Y, R, b) or simply \sigma(Y, R) (see footnote for details).The weak topology on Y is the weakest TVS topology on Y making all maps b(x, \,\cdot\,) : Y \to \mathbb continuous, as x ranges over R. The dual notation of (Y, \sigma(Y, R, b)), (Y, \sigma(Y, R)), or simply (Y, \sigma) may also be used to denote Y endowed with the weak topology \sigma(Y, R, b). If R is not clear from context then it should be assumed to be all of X, in which case it is simply called the weak topology on Y (induced by X). :: If "\sigma(X, Y, b)" is attached to a topological definition (e.g. \sigma(X, Y, b)-converges, \sigma(X, Y, b)-bounded, \operatorname_(S), etc.) then it means that definition when the first space (i.e. X) carries the \sigma(X, Y, b) topology. Mention of b or even X and Y may be omitted if no confusion arises. So, for instance, if a sequence \left(a_i\right)_^ in Y "\sigma-converges" or "weakly converges" then this means that it converges in (Y, \sigma(Y, X, b)) whereas if it were a sequence in X, then this would mean that it converges in (X, \sigma(X, Y, b))). The topology \sigma(X, Y, b) is locally convex since it is determined by the family of seminorms p_y : X \to \R defined by p_y(x) := , b(x, y), , as y ranges over Y. If x \in X and \left(x_i\right)_ is a net in X, then \left(x_i\right)_ \sigma(X, Y, b)-converges to x if \left(x_i\right)_ converges to x in (X, \sigma(X, Y, b)). A net \left(x_i\right)_ \sigma(X, Y, b)-converges to x if and only if for all y \in Y, b\left(x_i, y\right) converges to b(x, y). If \left(x_i\right)_^ is a sequence of orthonormal vectors in Hilbert space, then \left(x_i\right)_^ converges weakly to 0 but does not norm-converge to 0 (or any other vector). If (X, Y, b) is a pairing and N is a proper vector subspace of Y such that (X, N, b) is a dual pair, then \sigma(X, N, b) is strictly coarser than \sigma(X, Y, b).


Bounded subsets

A subset S of X is \sigma(X, Y, b)-bounded if and only if \sup_ , b(S, y), < \infty \quad \text y \in Y, where , b(S, y), := \.


Hausdorffness

If (X, Y, b) is a pairing then the following are equivalent: # X distinguishes points of Y; # The map y \mapsto b(\,\cdot\,, y) defines an injection from Y into the algebraic dual space of X; # \sigma(Y, X, b) is Hausdorff.


Weak representation theorem

The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of (X, \sigma(X, Y, b)). Consequently, the continuous dual space of (X, \sigma(X, Y, b)) is (X, \sigma(X, Y, b))^ = b(\,\cdot\,, Y) := \left\. With respect to the canonical pairing, if X is a TVS whose continuous dual space X^ separates points on X (i.e. such that \left(X, \sigma\left(X, X^\right)\right) is Hausdorff, which implies that X is also necessarily Hausdorff) then the continuous dual space of \left(X^, \sigma\left(X^, X\right)\right) is equal to the set of all "evaluation at a point x" maps as x ranges over X (i.e. the map that send x^ \in X^ to x^(x)). This is commonly written as \left(X^, \sigma\left(X^, X\right)\right)^ = X \qquad \text \qquad \left(X^_\right)^ = X. This very important fact is why results for polar topologies on continuous dual spaces, such as the strong dual topology \beta\left(X^, X\right) on X^ for example, can also often be applied to the original TVS X; for instance, X being identified with \left(X^_\right)^ means that the topology \beta\left(\left(X^_\right)^, X^_\right) on \left(X^_\right)^ can instead be thought of as a topology on X. Moreover, if X^ is endowed with a topology that is finer than \sigma\left(X^, X\right) then the continuous dual space of X^ will necessarily contain \left(X^_\right)^ as a subset. So for instance, when X^ is endowed with the strong dual topology (and so is denoted by X^_) then \left(X^_\right)^ ~\supseteq~ \left(X^_\right)^ ~=~ X which (among other things) allows for X to be endowed with the subspace topology induced on it by, say, the strong dual topology \beta\left(\left(X^_\right)^, X^_\right) (this topology is also called the strong bidual topology and it appears in the theory of
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
s: the Hausdorff locally convex TVS X is said to be if \left(X^_\right)^ = X and it will be called if in addition the strong bidual topology \beta\left(\left(X^_\right)^, X^_\right) on X is equal to X's original/starting topology).


Orthogonals, quotients, and subspaces

If (X, Y, b) is a pairing then for any subset S of X: If X is a normed space then under the canonical duality, S^ is norm closed in X^ and S^ is norm closed in X.


Subspaces

Suppose that M is a vector subspace of X and let (M, Y, b) denote the restriction of (X, Y, b) to M \times Y. The weak topology \sigma(M, Y, b) on M is identical to the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
that M inherits from (X, \sigma(X, Y, b)). Also, \left(M, Y / M^, b\big\vert_M\right) is a paired space (where Y / M^ means Y / \left(M^\right)) where b\big\vert_M : M \times Y / M^ \to \mathbb is defined by \left(m, y + M^\right) \mapsto b(m, y). The topology \sigma\left(M, Y / M^, b\big\vert_M\right) is equal to the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
that M inherits from (X, \sigma(X, Y, b)). Furthermore, if (X, \sigma(X, Y, b)) is a dual system then so is \left(M, Y / M^, b\big\vert_M\right).


Quotients

Suppose that M is a vector subspace of X. Then \left(X / M, M^, b / M\right) is a paired space where b / M : X / M \times M^ \to \mathbb is defined by (x + M, y) \mapsto b(x, y). The topology \sigma\left(X / M, M^\right) is identical to the usual
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
induced by (X, \sigma(X, Y, b)) on X / M.


Polars and the weak topology

If X is a locally convex space and if H is a subset of the continuous dual space X^, then H is \sigma\left(X^, X\right)-bounded if and only if H \subseteq B^ for some
barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
B in X. The following results are important for defining polar topologies. If (X, Y, b) is a pairing and A \subseteq X, then:
  1. The polar A^ of A is a closed subset of (Y, \sigma(Y, X, b)).
  2. The polars of the following sets are identical: (a) A; (b) the convex hull of A; (c) the balanced hull of A; (d) the \sigma(X, Y, b)-closure of A; (e) the \sigma(X, Y, b)-closure of the convex balanced hull of A.
  3. The
    bipolar theorem In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone ...
    : The bipolar of A, denoted by A^, is equal to the \sigma(X, Y, b)-closure of the convex balanced hull of A. * The
    bipolar theorem In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone ...
    in particular "is an indispensable tool in working with dualities."
  4. A is \sigma(X, Y, b)-bounded if and only if A^ is absorbing in Y.
  5. If in addition Y distinguishes points of X then A is \sigma(X, Y, b)- bounded if and only if it is \sigma(X, Y, b)- totally bounded.
If (X, Y, b) is a pairing and \tau is a locally convex topology on X that is consistent with duality, then a subset B of X is a
barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
in (X, \tau) if and only if B is the polar of some \sigma(Y, X, b)-bounded subset of Y.


Transposes


Transposes of a linear map with respect to pairings

Let (X, Y, b) and (W, Z, c) be pairings over \mathbb and let F : X \to W be a linear map. For all z \in Z, let c(F(\,\cdot\,), z) : X \to \mathbb be the map defined by x \mapsto c(F(x), z). It is said that Fs transpose or adjoint is well-defined if the following conditions are satisfied: # X distinguishes points of Y (or equivalently, the map y \mapsto b(\,\cdot\,, y) from Y into the algebraic dual X^ is
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
), and # c(F(\,\cdot\,), Z) \subseteq b(\,\cdot\,, Y), where c(F(\,\cdot\,), Z) := \ and b(\,\cdot\,, Y) := \. In this case, for any z \in Z there exists (by condition 2) a unique (by condition 1) y \in Y such that c(F(\,\cdot\,), z) = b(\,\cdot\,, y)), where this element of Y will be denoted by ^t F(z). This defines a linear map ^t F : Z \to Y called the transpose or adjoint of F with respect to (X, Y, b) and (W, Z, c) (this should not be confused with the
Hermitian adjoint In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \l ...
). It is easy to see that the two conditions mentioned above (i.e. for "the transpose is well-defined") are also necessary for ^t F to be well-defined. For every z \in Z, the defining condition for ^t F(z) is c(F(\,\cdot\,), z) = b\left(\,\cdot\,, ^t F(z)\right), that is, c(F(x), z) = b\left(x, ^t F(z)\right) for all x \in X. By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form Z \to Y,If G : Z \to Y is a linear map then G's transpose, ^t G : X \to W, is well-defined if and only if Z distinguishes points of W and b(X, G(\,\cdot\,)) \subseteq c(W, \,\cdot\,). In this case, for each x \in X, the defining condition for ^t G(x) is: c(x, G(\,\cdot\,)) = c\left(^t G(x), \,\cdot\,\right). X \to Z,If H : X \to Z is a linear map then H's transpose, ^t H : W \to Y, is well-defined if and only if X distinguishes points of Y and c(W, H(\,\cdot\,)) \subseteq b(\,\cdot\,, Y). In this case, for each w \in W, the defining condition for ^t H(w) is: c(w, H(\,\cdot\,)) = b\left(\,\cdot\,, ^t H(w)\right). W \to Y,If H : W \to Y is a linear map then H's transpose, ^t H : X \to Q, is well-defined if and only if W distinguishes points of Z and b(X, H(\,\cdot\,)) \subseteq c(\,\cdot\,, Z). In this case, for each x \in X, the defining condition for ^t H(x) is: c(x, H(\,\cdot\,)) = b\left(\,\cdot\,, ^t H(x)\right). Y \to W,If H : Y \to W is a linear map then H's transpose, ^t H : Z \to X, is well-defined if and only if Y distinguishes points of X and c(H(\,\cdot\,), Z) \subseteq b(X, \,\cdot\,). In this case, for each z \in Z, the defining condition for ^t H(z) is: c(H(\,\cdot\,), z) = b\left(^t H(z), \,\cdot\,\right) etc. (see footnote).


Properties of the transpose

Throughout, (X, Y, b) and (W, Z, c) be pairings over \mathbb and F : X \to W will be a linear map whose transpose ^t F : Z \to Y is well-defined. * ^t F : Z \to Y is
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
(i.e. \operatorname ^t F = \) if and only if the range of F is dense in \left(W, \sigma\left(W, Z, c\right)\right). * If in addition to ^t F being well-defined, the transpose of ^t F is also well-defined then ^ F = F. * Suppose (U, V, a) is a pairing over \mathbb and E : U \to X is a linear map whose transpose ^t E : Y \to V is well-defined. Then the transpose of F \circ E : U \to W, which is ^t (F \circ E) : Z \to V, is well-defined and ^t (F \circ E) = ^t E \circ ^t F. * If F : X \to W is a vector space isomorphism then ^t F : Z \to Y is bijective, the transpose of F^ : W \to X, which is ^t \left(F^\right) : Y \to Z, is well-defined, and ^t \left(F^\right) = \left(^t F\right)^ * Let S \subseteq X and let S^ denotes the absolute polar of A, then: *# (S) = \left(^t F\right)^\left(S^\right); *# if F(S) \subseteq T for some T \subseteq W, then ^t F\left(T^\right) \subseteq S^; *# if T \subseteq W is such that ^t F\left(T^\right) \subseteq S^, then F(S) \subseteq T^; *# if T \subseteq W and S \subseteq X are weakly closed disks then ^t F\left(T^\right) \subseteq S^ if and only if F(S) \subseteq T; *# \operatorname ^t F = F(X) . : These results hold when the real polar is used in place of the absolute polar. If X and Y are normed spaces under their canonical dualities and if F : X \to Y is a continuous linear map, then \, F\, = \left\, ^t F\right\, .


Weak continuity

A linear map F : X \to W is weakly continuous (with respect to (X, Y, b) and (W, Z, c)) if F : (X, \sigma(X, Y, b)) \to (W, (W, Z, c)) is continuous. The following result shows that the existence of the transpose map is intimately tied to the weak topology.


Weak topology and the canonical duality

Suppose that X is a vector space and that X^ is its the algebraic dual. Then every \sigma\left(X, X^\right)-bounded subset of X is contained in a finite dimensional vector subspace and every vector subspace of X is \sigma\left(X, X^\right)-closed.


Weak completeness

If (X, \sigma(X, Y, b)) is a
complete topological vector space In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
say that X is \sigma(X, Y, b)-complete or (if no ambiguity can arise) weakly-complete. There exist
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s that are not weakly-complete (despite being complete in their norm topology). If X is a vector space then under the canonical duality, \left(X^, \sigma\left(X^, X\right)\right) is complete. Conversely, if Z is a Hausdorff locally convex TVS with continuous dual space Z^, then \left(Z, \sigma\left(Z, Z^\right)\right) is complete if and only if Z = \left(Z^\right)^; that is, if and only if the map Z \to \left(Z^\right)^ defined by sending z \in Z to the evaluation map at z (i.e. z^ \mapsto z^(z)) is a bijection. In particular, with respect to the canonical duality, if Y is a vector subspace of X^ such that Y separates points of X, then (Y, \sigma(Y, X)) is complete if and only if Y = X^. Said differently, there does exist a proper vector subspace Y \neq X^ of X^ such that (X, \sigma(X, Y)) is Hausdorff and Y is complete in the weak-* topology (i.e. the topology of pointwise convergence). Consequently, when the continuous dual space X^ of a Hausdorff locally convex TVS X is endowed with the weak-* topology, then X^_ is complete if and only if X^ = X^ (that is, if and only if linear functional on X is continuous).


Identification of ''Y'' with a subspace of the algebraic dual

If X distinguishes points of Y and if Z denotes the range of the injection y \mapsto b(\,\cdot\,, y) then Z is a vector subspace of the algebraic dual space of X and the pairing (X, Y, b) becomes canonically identified with the canonical pairing \langle X, Z \rangle (where \left\langle x, x^ \right\rangle := x^(x) is the natural evaluation map). In particular, in this situation it will be assumed without loss of generality that Y is a vector subspace of X's algebraic dual and b is the evaluation map. :: Often, whenever y \mapsto b(\,\cdot\,, y) is injective (especially when (X, Y, b) forms a dual pair) then it is common practice to assume without loss of generality that Y is a vector subspace of the algebraic dual space of X, that b is the natural evaluation map, and also denote Y by X^. In a completely analogous manner, if Y distinguishes points of X then it is possible for X to be identified as a vector subspace of Y's algebraic dual space.


Algebraic adjoint

In the special case where the dualities are the canonical dualities \left\langle X, X^ \right\rangle and \left\langle W, W^ \right\rangle, the transpose of a linear map F : X \to W is always well-defined. This transpose is called the algebraic adjoint of F and it will be denoted by F^; that is, F^ = ^t F : W^ \to X^. In this case, for all w^ \in W^, F^\left(w^\right) = w^ \circ F where the defining condition for F^\left(w^\right) is: \left\langle x, F^\left(w^\right) \right\rangle = \left\langle F(x), w^ \right\rangle \quad \text >x \in X, or equivalently, F^\left(w^\right)(x) = w^(F(x)) \quad \text x \in X. If X = Y = \mathbb^n for some integer n, \mathcal = \left\ is a basis for X with
dual basis In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimension of V), the dual set of B is a set B^* of vectors in the dual space V^* with the same index set I such that B and ...
\mathcal^ = \left\, F : \mathbb^n \to \mathbb^n is a linear operator, and the matrix representation of F with respect to \mathcal is M := \left(f_\right), then the transpose of M is the matrix representation with respect to \mathcal^ of F^.


Weak continuity and openness

Suppose that \left\langle X, Y \right\rangle and \langle W, Z \rangle are canonical pairings (so Y \subseteq X^ and Z \subseteq W^) that are dual systems and let F : X \to W be a linear map. Then F : X \to W is weakly continuous if and only if it satisfies any of the following equivalent conditions: # F : (X, \sigma(X, Y)) \to (W, \sigma(W, Z)) is continuous. # F^(Z) \subseteq Y # the transpose of ''F'', ^t F : Z \to Y, with respect to \left\langle X, Y \right\rangle and \langle W, Z \rangle is well-defined. If F is weakly continuous then ^t F : : (Z, \sigma(Z, W)) \to (Y, \sigma(Y, X)) will be continuous and furthermore, ^ F = F A map g : A \to B between topological spaces is relatively open if g : A \to \operatorname g is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
ping, where \operatorname g is the range of g. Suppose that \langle X, Y \rangle and \langle W, Z \rangle are dual systems and F : X \to W is a weakly continuous linear map. Then the following are equivalent: # F : (X, \sigma(X, Y)) \to (W, \sigma(W, Z)) is relatively open. # The range of ^t F is \sigma(Y, X)-closed in Y; # \operatorname ^t F = (\operatorname F)^ Furthermore, * F : X \to W is injective (resp. bijective) if and only if ^t F is surjective (resp. bijective); * F : X \to W is surjective if and only if ^t F : : (Z, \sigma(Z, W)) \to (Y, \sigma(Y, X)) is relatively open and injective.


=Transpose of a map between TVSs

= The transpose of map between two TVSs is defined if and only if F is weakly continuous. If F : X \to Y is a linear map between two Hausdorff locally convex topological vector spaces, then: * If F is continuous then it is weakly continuous and ^t F is both Mackey continuous and strongly continuous. * If F is weakly continuous then it is both Mackey continuous and strongly continuous (defined below). * If F is weakly continuous then it is continuous if and only if ^t F : ^ \to X^ maps equicontinuous subsets of Y^ to equicontinuous subsets of X^. * If X and Y are normed spaces then F is continuous if and only if it is weakly continuous, in which case \, F\, = \left\, ^t F\right\, . * If F is continuous then F : X \to Y is relatively open if and only if F is weakly relatively open (i.e. F : \left(X, \sigma\left(X, X^\right)\right) \to \left(Y, \sigma\left(Y, Y^\right)\right) is relatively open) and every equicontinuous subsets of \operatorname ^t F = ^t F\left(Y^\right) is the image of some equicontinuous subsets of Y^. * If F is continuous injection then F : X \to Y is a TVS-embedding (or equivalently, a
topological embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...
) if and only if every equicontinuous subsets of X^ is the image of some equicontinuous subsets of Y^.


Metrizability and separability

Let X be a locally convex space with continuous dual space X^ and let K \subseteq X^. # If K is equicontinuous or \sigma\left(X^, X\right)-compact, and if D \subseteq X^ is such that \operatorname D is dense in X, then the subspace topology that K inherits from \left(X^, \sigma\left(X^, D\right)\right) is identical to the subspace topology that K inherits from \left(X^, \sigma\left(X^, X\right)\right). # If X is separable and K is equicontinuous then K, when endowed with the subspace topology induced by \left(X^, \sigma\left(X^, X\right)\right), is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
. # If X is separable and
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
, then \left(X^, \sigma\left(X^, X\right)\right) is separable. # If X is a normed space then X is separable if and only if the closed unit call the continuous dual space of X is metrizable when given the subspace topology induced by \left(X^, \sigma\left(X^, X\right)\right). # If X is a normed space whose continuous dual space is separable (when given the usual norm topology), then X is separable.


Polar topologies and topologies compatible with pairing

Starting with only the weak topology, the use of
polar set In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X, lying in the dual space X^. The bipolar of a subset is the polar of A^\circ, but ...
s produces a range of locally convex topologies. Such topologies are called polar topologies. The weak topology is the weakest topology of this range. Throughout, (X, Y, b) will be a pairing over \mathbb and \mathcal will be a non-empty collection of \sigma(X, Y, b)-bounded subsets of X.


Polar topologies

Given a collection \mathcal of subsets of X, the polar topology on Y determined by \mathcal (and b) or the \mathcal-topology on Y is the unique
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) topology on Y for which \left\ forms a subbasis of neighborhoods at the origin. When Y is endowed with this \mathcal-topology then it is denoted by ''Y''\mathcal. Every polar topology is necessarily locally convex. When \mathcal is a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
with respect to subset inclusion (i.e. if for all G, K \in \mathcal there exists some K \in \mathcal such that G \cup H \subseteq K) then this neighborhood subbasis at 0 actually forms a neighborhood basis at 0. The following table lists some of the more important polar topologies. :: If \Delta(X, Y, b) denotes a polar topology on Y then Y endowed with this topology will be denoted by Y_, Y_ or simply Y_ (e.g. for \sigma(Y, X, b) we'd have \Delta = \sigma so that Y_, Y_ and Y_ all denote Y endowed with \sigma(X, Y, b)).


Definitions involving polar topologies

Continuity A linear map F : X \to W is Mackey continuous (with respect to (X, Y, b) and (W, Z, c)) if F : (X, \tau(X, Y, b)) \to (W, \tau(W, Z, c)) is continuous. A linear map F : X \to W is strongly continuous (with respect to (X, Y, b) and (W, Z, c)) if F : (X, \beta(X, Y, b)) \to (W, \beta(W, Z, c)) is continuous.


Bounded subsets

A subset of X is weakly bounded (resp. Mackey bounded, strongly bounded) if it is bounded in (X, \sigma(X, Y, b)) (resp. bounded in (X, \tau(X, Y, b)), bounded in (X, \beta(X, Y, b))).


Topologies compatible with a pair

If (X, Y, b) is a pairing over \mathbb and \mathcal is a vector topology on X then \mathcal is a topology of the pairing and that it is compatible (or consistent) with the pairing (X, Y, b) if it is locally convex and if the continuous dual space of \left(X, \mathcal\right) = b(\,\cdot\,, Y).Of course, there is an analogous definition for topologies on Y to be "compatible it a pairing" but this article will only deal with topologies on X. If X distinguishes points of Y then by identifying Y as a vector subspace of X's algebraic dual, the defining condition becomes: \left(X, \mathcal\right)^ = Y. Some authors (e.g. rèves 2006and chaefer 1999 require that a topology of a pair also be Hausdorff, which it would have to be if Y distinguishes the points of X (which these authors assume). The weak topology \sigma(X, Y, b) is compatible with the pairing (X, Y, b) (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 In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not ...
. If N is a normed space that is not reflexive then the usual norm topology on its continuous dual space is compatible with the duality \left(N^, N\right).


Mackey–Arens theorem

The following is one of the most important theorems in duality theory. It follows that the Mackey topology \tau(X, Y, b), which recall is the polar topology generated by all \sigma(X, Y, b)-compact disks in Y, is the strongest locally convex topology on X that is compatible with the pairing (X, Y, b). 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's theorem, barrels, and closed convex sets

If X is a TVS (over \Reals or \Complex) then a half-space is a set of the form \ for some real r and some continuous linear functional f on X. The above theorem implies that the closed and convex subsets of a locally convex space depend on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality;that is, if \mathcal and \mathcal are any locally convex topologies on X with the same continuous dual spaces, then a convex subset of X is closed in the \mathcal topology if and only if it is closed in the \mathcal topology. This implies that the \mathcal-closure of any convex subset of X is equal to its \mathcal-closure and that for any \mathcal-closed disk A in X, A = A^. In particular, if B is a subset of X then B is a
barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
in (X, \mathcal) if and only if it is a barrel in (X, \mathcal). The following theorem shows that barrels (i.e. closed absorbing disks) are exactly the polars of weakly bounded subsets. If X is a topological vector space, then: # A closed absorbing and balanced subset B of X absorbs each convex compact subset of X (i.e. there exists a real r > 0 such that r B contains that set). # If X is Hausdorff and locally convex then every barrel in X absorbs every convex bounded complete subset of X. 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.


Space of finite sequences

Let X denote the space of all sequences of scalars r_ = \left(r_i\right)_^ such that r_i = 0 for all sufficiently large i. Let Y = X and define a bilinear map b : X \times X \to \mathbb by b\left(r_, s_\right) := \sum_^ r_i s_i. Then \sigma(X, X, b) = \tau(X, X, b). Moreover, a subset T \subseteq X is \sigma(X, X, b)-bounded (resp. \beta(X, X, b)-bounded) if and only if there exists a sequence m_ = \left(m_i\right)_^ of positive real numbers such that \left, t_i\ \leq m_i for all t_ = \left(t_i\right)_^ \in T and all indices i (resp. and m_ \in X). It follows that there are weakly bounded (that is, \sigma(X, X, b)-bounded) subsets of X that are not strongly bounded (that is, not \beta(X, X, b)-bounded).


See also

* * * * * * * * * * * * * * *


Notes


References


Bibliography

* * Michael Reed and Barry Simon, ''Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis,'' Section III.3. Academic Press, San Diego, 1980. . * * * *


External links


Duality Theory
{{DEFAULTSORT:Dual Pair Functional analysis Pair Linear functionals Topological vector spaces