Dual system

In the field of functional analysis, a subfield of mathematics, a dual system, dual pair, or a duality over a field 𝕂 (𝕂 is either the real or the complex numbers) is a triple (X, Y, b) consisting of two vector spaces over 𝕂 and a bilinear map b : X × Y → 𝕂 such that for all non-zero x ∈ X the map y ↦ b(x, y) is not identically 0 and for all non-zero y ∈ Y, the map x ↦ b(x, y) is not identically 0. The study of dual systems is called duality theory.

This article is about dual pairs of vector spaces. For dual pairs in representation theory, see Reductive dual pair.

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

A pairing or a pair over a field 𝕂 is a triple (X, Y, b), which may also be denoted by b(X, Y), consisting of two vector spaces X and Y over 𝕂 (which this article assumes is either the real numbers ℝ or the complex numbers ℂ) and a bilinear map b : X × Y → 𝕂, which we call the bilinear map associated with the pairing[2] or simply the pairing's map/bilinear form.

Notation: For all x ∈ X, let b(x, •) : Y → 𝕂 denote the linear functional on Y defined by y ↦ b(x, y) and let b(X, •) := { b(x, •) : x ∈ X} . Similarly, for all y ∈ Y, let b(•, y) : X → 𝕂 be defined by x ↦ b(x, y) and let b(•, Y) := { b(•, y) : y ∈ Y} .

Notation: It is common practice to write ⟨x, y⟩ instead of b(x, y), in which case the pair is often denoted by ⟨X, Y⟩ rather than (X, Y, ⟨·, ·⟩).
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.

A pairing (X, Y, b) is called a dual system, a dual pair,[3] or a duality over 𝕂 if it satisfies the following two separation axioms:

1. Y separates/distinguishes points of X: if x ∈ X is such that b(x, •) = 0 then x = 0; or equivalently, for all non-zero x ∈ X, the map b(x, •) : Y → 𝕂 is not identically 0 (i.e. there exists a y ∈ Y such that b(x, y) ≠ 0);
2. X separates/distinguishes points of Y: if y ∈ Y is such that b(•, y) = 0 then y = 0; or equivalently, for all non-zero y ∈ Y, the map b(•, y) : X → 𝕂 is not identically 0 (i.e. there exists an x ∈ X such that b(x, y) ≠ 0).

In this case we say that b is non-degenerate, we say that b places X and Y in duality (or in separated duality), and we call b the duality pairing of the (X, Y, b).[2][3]

A subset S of Y is called total if for all x ∈ X, b(x, s) = 0 for all s ∈ S implies x = 0. A total subset of X is defined analogously (see footnote).[4]

We say that the elements x ∈ X and y ∈ Y are orthogonal and write x ⟂ y if b(x, y) = 0. We say that two sets R ⊆ X and S ⊆ Y are orthogonal and write R ⟂ S if r and s are orthogonal for all r ∈ R and s ∈ S and we say that R is orthogonal to an element y ∈ Y if R is orthogonal to { y } . For R ⊆ X, we define the orthogonal or annihilator of R to be R^⟂ := { y ∈ Y : R ⟂ y } := { y ∈ Y : b(R, y) = { 0 } } .

Polar sets

Main article: Polar set

Throughout, (X, Y, b) will be a pairing over 𝕂. The absolute polar or polar of a subset A of X is the set:[5]

${\displaystyle A^{\circ }:=\left\{y\in Y:\sup _{x\in A}\left|b(x,y)\right|\leq 1\right\}}$.

Dually, the absolute polar or polar of a subset B of Y is denoted by B° and defined by

${\displaystyle B^{\circ }:=\left\{x\in X:\sup _{y\in B}\left|b(x,y)\right|\leq 1\right\}}$.

In this case, the absolute polar of a subset B of Y is also called the absolute prepolar or prepolar of B and may be denoted by °B.

The polar B° is necessarily a convex set containing 0 ∈ 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.[6]

If A ⊆ X then the bipolar of A, denoted by A°°, is the set °(A^⟂). Similarly, if B ⊆ Y then the bipolar of B is B°° := (°B)°.

If A is a vector subspace of X, then A° = A^⟂ and this is also equal to the real polar of A.

Dual definitions and results

Given a pairing (X, Y, b) we can define a new pairing (Y, X, d) where d(y, x) := b(x, y) for all x ∈ X and all y ∈ Y.[2] For the purpose of explaining the following convention, we will call d the mirror of b (this is not standard terminology).

There is a repeating theme in duality theory, which is that any definition for a pairing (X, Y, b) has a corresponding dual definition for the pairing (Y, X, d).

Convention and Definition: Given any definition for a pairing (X, Y, b), one obtains a dual definition by applying it to the pairing (Y, X, d). This conventions also apply to theorems.

Convention: Adhering to common practice, unless clarity is needed, whenever we give a definition (or result) for a pairing (X, Y, b) then we will omit mention of the corresponding dual definition (or result) but nevertheless use it.

For instance, if we define "X distinguishes points of Y" (resp, "S is a total subset of Y") as above, then under this convention we immediately obtain the dual definition of "Y distinguishes points of X" (resp, "S is a total subset of X").

This following notation is almost ubiquitous because it allows us to avoid having to assign a symbol to the mirror of b.

Convention and Notation: If a definition and its notation for a pairing (X, Y, b) depends on the order of X and Y (e.g. the definition of the Mackey topology 𝜏(X, Y, b) on X) then if we switch the order of X and Y then we mean that definition applied to (Y, X, d) (e.g. 𝜏(Y, X, b) actually denotes the topology 𝜏(Y, X, d)).

For instance, once we define the weak topology on X, which is denoted by 𝜎(X, Y, b), then we will automatically apply this definition to the pairing (Y, X, d) so as to obtain the definition of the weak topology on Y, where we will denote the this topology 𝜎(Y, X, b) rather than 𝜎(Y, X, d).

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

Although it is technically incorrect and an abuse of notation, we will also adhere to the following nearly ubiquitous convention:

Convention: This article will use the common practice of treating a pairing (X, Y, b) interchangeably with (Y, X, d) and also 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 × N is the pairing ${\displaystyle \left(M,N,b{\big \vert }_{M\times N}\right)}$. Note that if (X, Y, b) is a duality then it's possible for a restrictions to fail to be a duality (e.g. if Y ≠ { 0 } and N = { 0 } ).

Convention: We will use the common practice of denoting the restriction ${\displaystyle \left(M,N,b{\big \vert }_{M\times N}\right)}$ 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 (i.e. the space of all linear functionals on X). There is a canonical duality (X, X^#, c) where c(x, x') = ⟨x, x'⟩ := x'(x), which is called the evaluation map or the natural or canonical bilinear functional on X × X^#. Note in particular that for any x' ∈ X^#, c(•, x') is just another way of denoting x'; i.e. c(•, x') = x'(•) = x'.

If N is a vector subspace of X^# then the restriction of (X, X^#, c) to X × N is called the canonical pairing where if this pairing is a duality then we will instead call it 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.

Notation: The evaluation map will be denoted by ⟨x, x'⟩ := x'(x) (rather than by c) and we will write ⟨X, N⟩ 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, we will assume that they are associated with the canonical pairing ⟨X, N⟩.

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 (i.e. n(x) = 0 for all n ∈ N implies x = 0).[2]

Canonical duality on a topological vector space

Suppose X is a topological vector space (TVS) with continuous dual space X'. Then the restriction of the canonical duality ${\displaystyle \left(X,X^{\#},c\right)}$ to X × X' defines a pairing ${\displaystyle \left(X,X^{\prime },c{\big \vert }_{X\times X^{\prime }}\right)}$ 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.[3]

Assumption: As is commonly done, whenever X is a TVS then, unless indicated otherwise, we will assume without comment that its associated with the canonical pairing ⟨X, X'⟩.

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 X be a TVS with algebraic dual X^# and let 𝒩 be a basis of neighborhoods of X at the origin. Under the canonical duality ⟨X, X^#⟩, the continuous dual space of X is the union of all N° as N ranges over 𝒩 (where the polars are taken in X^#).

Other examples

• If (H, ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$) is a complex Hilbert space then ${\displaystyle \left(H,H,\left\langle \cdot ,\cdot \right\rangle \right)}$ forms a dual system if and only if dim H = 0. If H is non-trivial then ${\displaystyle \left(H,H,\left\langle \cdot ,\cdot \right\rangle \right)}$ doesn't even form pairing since the inner product is sesquilinear rather than bilinear.[2]
• If (H, ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$) is a real Hilbert space then ${\displaystyle \left(H,H,\left\langle \cdot ,\cdot \right\rangle \right)}$ forms a dual system.
• Suppose X = ℝ², Y = ℝ³, and ${\displaystyle b\left(\left(x_{1},y_{1}\right),\left(x_{1},y_{1},z_{1}\right)\right):=x_{1}x_{2}+y_{1}y_{2}}$ for all ${\displaystyle \left(x_{1},y_{1}\right)}$ ∈ X and ${\displaystyle \left(x_{1},y_{1},z_{1}\right)}$ ∈ Y. Then (X, Y, b) is a pairing such that X distinguishes points of Y, but Y does not distinguish points of X. Furthermore, X^⟂ := { y ∈ Y : X ⟂ y } = { (0, 0, z) : z ∈ ℝ }
• Let 1 < p < ∞, X := L^p(μ), Y := L^q(μ) (where q is such that  1/p + 1/q = 1), and b(f, g) := ${\displaystyle \int fg\,\mathrm {d} \mu }$. Then (X, Y, b) is a dual system.
• Let X and Y be vector spaces over the same field 𝕂. Then the bilinear form ${\displaystyle b\left(x\otimes y,x^{*}\otimes y^{*}\right)=\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime },y\right\rangle }$ places X × Y and ${\displaystyle X^{\#}\times Y^{\#}}$ in duality.[3]
• A sequence space X and its beta dual ${\displaystyle Y:=X^{\beta }}$ with the bilinear map defined as ${\displaystyle \langle x,y\rangle$ :=\sum _{i=1}^{\infty }x_{i}y_{i}} for x ∈ X, y ∈ ${\displaystyle X^{\beta }}$ forms a dual system.

Weak topology

Main articles: Weak topology and Weak-* topology

Suppose that (X, Y, b) is a pairing of vector spaces over 𝕂. If S ⊆ Y then the weak topology on X induced by S (and b) is the weakest TVS topology on X, denoted by 𝜎(X, S, b) or simply 𝜎(X, S), making all maps b(•, y) : X → 𝕂 continuous, as y ranges over S.[2] We use X_{𝜎(X, S, b)}, X_{𝜎(X, S)}, or (if no confusion could arise) simply X_𝜎 to denote X endowed with the weak topology 𝜎(X, S, b). If we do not indicate what the subset S is, then by the weak topology on X we mean the weak topology on X induced by Y. Similarly, if R ⊆ X then we have the dual definition of the weak topology on Y induced by R (and b), which is denoted by 𝜎(Y, R, b) or simply 𝜎(Y, R) (see footnote for details).[7] Observe that the weak topology depends entirely on the function b and the usual topology on ℂ (the topology doesn't even depend on the algebraic structures of X and Y).

Definition and notation: If we attach "𝜎(X, Y, b)" to a topological definition (e.g. 𝜎(X, Y, b)-converges, 𝜎(X, Y, b)-bounded, cl_{𝜎(X, Y, b)}(S), etc.) then we mean that definition when the first space (i.e. X) carries the 𝜎(X, Y, b) topology. We may also omit mentioning b or even X and Y if no confusion will arise. So for instance, if we say that a sequence ${\displaystyle \left(a_{i}\right)_{i=1}^{\infty }}$ in Y "𝜎-converges" or "weakly converges" then this means that it converges in (Y, 𝜎(Y, X, b)) whereas if it were a sequence in X then this would mean that it converges in (X, 𝜎(X, Y, b)).

The topology 𝜎(X, Y, b) is locally convex since it is determined by the family of seminorms p_y : X → ℝ defined by p_y(x) := |b(x, y)|, as y ranges over Y.[2] If x ∈ X and ${\displaystyle \left(x_{i}\right)_{i\in I}}$ is a net in X, then we say that ${\displaystyle \left(x_{i}\right)_{i\in I}}$ 𝜎(X, Y, b)-converges to x if ${\displaystyle \left(x_{i}\right)_{i\in I}}$ converges to X in (X, 𝜎(X, Y, b)).[2] A net ${\displaystyle \left(x_{i}\right)_{i\in I}}$ 𝜎(X, Y, b)-converges to x if and only if for all y ∈ Y, b(x_i, y) converges to b(x, y). Note that if ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ is a sequence of orthonormal vectors in Hilbert space, then ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ converges weakly to 0 but does not norm-converge to 0 (or any other vector).[2]

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 𝜎(X, N, b) is strictly coarser than 𝜎(X, Y, b).[2]

Bounded subsets

A subset S of X is 𝜎(X, Y, b)-bounded if and only if sup |b(S, y)| < ∞ for all y ∈ Y, where |b(S, y)| := {​|b(s, y)| : s ∈ S} .

Hausdorffness

Proposition — If (X, Y, b) is a pairing then the following are equivalent:

1. X distinguishes points of Y;
2. The map y ↦ b(•, y) defines an injection from Y into the algebraic dual space of X;[2]
3. 𝜎(Y, X, b) is Hausdorff.[2]

Weak representation theorem

The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of (X, 𝜎(X, Y, b)).

Weak representation theorem[2] — Let (X, Y, b) be a pairing over the field 𝕂. Then the continuous dual space of (X, 𝜎(X, Y, b)) is b(•, Y) := { b(•, y) : y ∈ Y} . Furthermore,

1. If f is a continuous linear functional on (X, 𝜎(X, Y, b)) then there exists some y ∈ Y such that f = b(•, y); if such a y exists then it is unique if and only if X distinguishes points of Y.
2. The continuous dual space of (X, 𝜎(X, Y, b)) may be identified with Y/X^⟂, where X^⟂ := { y ∈ Y : b(x, y) = 0 for all x ∈ X } .

   Note that this is true regardless of whether or not X distinguishes points of Y or Y distinguishes points of X.

Orthogonals, quotients, and subspaces

If (X, Y, b) is a pairing then for any subset S of X:

• S^⟂ = (span S)^⟂ = (Cl_{𝜎(Y, X, b)}(span S))^⟂ = S^⟂⟂⟂ and this set is 𝜎(Y, X, b)-closed;[2]
• S ⊆ S^⟂⟂ = Cl_{𝜎(X, Y, b)}(span S);[2]
  • Thus if S is a 𝜎(X, Y, b)-closed vector subspce of X then S = S^⟂⟂.
• If (S_i)_{i ∈ I} is a family of 𝜎(X, Y, b)-closed vector subspaces of X then (∩_{i ∈ I} S_i)^⟂ = Cl_{𝜎(Y, X, b)}(span (∪_{i ∈ I} S^⟂
  _i)).[2]
• If (S_i)_{i ∈ I} is a family of subsets of X then (∪_{i ∈ I} S_i)^⟂ = ∩_{i ∈ I} S^⟂
  _i.[2]

If X is a normed space then under the canonical duality, S^⟂ is norm closed in X ' and S^⟂⟂ is norm closed in X.[2]

Subspaces

Suppose that M is a vector subspace of X and let (M, Y, b) denote the restriction of (X, Y, b) to M × Y. The weak topology 𝜎(M, Y, b) on M is identical to the subspace topology that M inherits from (X, 𝜎(X, Y, b)).

Also, (M, Y/M^⟂, b|_M) is a paired space (where Y/M^⟂ means Y/(M^⟂)) where b|_M : M × Y/M^⟂ → 𝕂 is defined by

(m, y + M^⟂) ↦ b(m, y).

The topology 𝜎(M, Y/M^⟂, b|_M) is equal to the subspace topology that M inherits from (X, 𝜎(X, Y, b)).[8] Furthermore, if (X, 𝜎(X, Y, b)) is a dual system then so is (M, Y/M^⟂, b|_M).[8]

Quotients

Suppose that M is a vector subspace of X. Then (X/M, M^⟂, b/M) is a paired space where b/M : X/M × M^⟂ → 𝕂 is defined by

(x + M, y) ↦ b(x, y).

The topology 𝜎(X/M, M^⟂) is identical to the usual quotient topology induced by (X, 𝜎(X, Y, b)) on X/M.[8]

Polars and the weak topology

Theorem[2] — If X is a locally convex space and if H is a subset of the continuous dual space X', then H is 𝜎(X', X)-bounded if and only if H ⊆ B° for some barrel B in X.

The following results are important for defining polar topologies. The bipolar theorem in particular "is an indispensable tool in working with dualities."[6]

Theorem[2] — Let (X, Y, b) be a pairing and A ⊆ X. Then

1. The polar of A, A°, is a closed subset of (Y, 𝜎(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 𝜎(X, Y, b)-closure of A; (e) the 𝜎(X, Y, b)-closure of the convex balanced hull of A.
3. The bipolar theorem: The bipolar of A, A°°, is equal to the 𝜎(X, Y, b)-closure of the convex balanced hull of A.
4. A is 𝜎(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 𝜎(X, Y, b)-bounded if and only if it is 𝜎(X, Y, b)-totally bounded.

Theorem[9] — Let (X, Y, b) be a pairing and let 𝜏 be a locally convex topology on X consistent with duality. Then a subset B of X is a barrel in (X, 𝜏) if and only if B is the polar of some 𝜎(Y, X, b)-bounded subset of Y.

Transposes

Transpose of a linear map with respect to pairings

See also: Transpose of a linear map, Transpose, and Transpose § Transposes of linear maps and bilinear forms

Let (X, Y, b) and (W, Z, c) be pairings over 𝕂 and let F : X → W be a linear map.

For all z ∈ Z, let c(F(•), z) : X → 𝕂 be the map defined by x ↦ c(F(x), z). We will say that F's transpose or adjoint is well-defined if the following conditions are satisfies:

1. X distinguishes points of Y (or equivalently, the map y ↦ b(•, y) from Y into the algebraic dual X^# is injective), and
2. c(F(•), Z) ⊆ b(•, Y), where c(F(•), Z) := { c(F(•), z) : z ∈ Z } .

In this case, for any z ∈ Z there exists (by condition 2) a unique (by condition 1) y ∈ Y such that c(F(•), z) = b(•, y), where we will denote by this element of Y by ^tF(z). This defines a linear map

^tF : Z → Y

called the transpose of adjoint of F with respect to (X, Y, b) and (W, Z, c) (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 ^tF to be well-defined. Note that for every z ∈ Z, the defining condition for ^tF(z) is

c(F(•), z) = b(•, ^tF(z)),

that is,

c(F(x), z) = b(x, ^tF(z))      for all x ∈ X.

Note that by the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form Z → Y,[10] X → Z,[11] W → Y,[12] Y → W,[13] etc. (see footnote for details).

Properties of the transpose

Throughout, (X, Y, b) and (W, Z, c) be pairings over 𝕂 and F : X → W will be a linear map whose transpose ^tF : Z → Y is well-defined.

• ^tF : Z → Y is injective (i.e. ker ^tF = { 0 } ) if and only if the range of F is dense in (W, 𝜎(W, Z, c)).[2]
• If in addition to ^tF being well-defined, the transpose of ^tF is also well-defined then ^{t t}F = F.
• Suppose (U, V, a) is a pairing over 𝕂 and E : U → X is a linear map whose transpose ^tE : Y → V is well-defined. Then the transpose of F ∘ E : U → W, which is ^t(F ∘ E) : Z → V, is well-defined and ^t(F ∘ E) = ^tE ∘ ^tF.
• If F : X → W is a vector space isomorphism then ^tF : Z → Y is bijective, the transpose of F⁻¹ : W → X, which is ^t(F⁻¹) : Y → Z, is well-defined, and ^t(F⁻¹) = (^tF)⁻¹.[2]
• Let S ⊆ X and let S° denotes the absolute polar of A, then:[2]
  1. [F(S)]° = (^tF)⁻¹(S°);
  2. if F(S) ⊆ T for some T ⊆ W, then ^tF(T°) ⊆ S°;
  3. if T ⊆ W is such that ^tF(T°) ⊆ S°, then F(S) ⊆ T°°;
  4. if T ⊆ W and S ⊆ X are weakly closed disks then ^tF(T°) ⊆ S° if and only if F(S) ⊆ T;
  5. ker ^tF = [Im F]° = [Im F]^⟂, where Im F := F(X) is the image of F.

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 → Y is a continuous linear map, then ||F|| = ||^tF||.[2]

Weak continuity

Definition: We say that the linear map F : X → W is weakly continuous (with respect to (X, Y, b) and (W, Z, c)) if F : (X, 𝜎(X, Y, b)) → (W, 𝜎(W, Z, c)) is continuous.

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

Proposition — Assume that X distinguishes points of Y and F : X → W is a linear map. Then the following are equivalent:

1. F is weakly continuous (i.e. F : (X, 𝜎(X, Y, b)) → (W, 𝜎(W, Z, c)) is continuous);
2. c(F(•), Z) ⊆ b(•, Y);
3. the transpose of F is well-defined.

If F is weakly continuous then

• ^tF : Z → Y is weakly continuous (i.e. ^tF : (Z, 𝜎(Z, W, c)) → (Y, 𝜎(Y, X, b)) is continuous);
• the transpose of ^tF is well-defined if and only if Z distinguishes points of W, in which case ^{t t}F = F.

Weak topology and the canonical duality

Suppose that X is a vector space and that X^# is its the algebraic dual. Then every 𝜎(X, X^#)-bounded subset of X is contained in a finite dimensional vector subspace and every vector subspace of X is 𝜎(X, X^#)-closed.[2]

Weak completeness

We call X 𝜎(X, Y, b)-complete or (if no ambiguity can arise) weakly-complete if (X, 𝜎(X, Y, b)) is a complete vector space. Note that there exist Banach spaces that are not weakly-complete (despite being complete).[2]

Theorem[2] — If X is a vector space then under the canonical duality, (X^#, 𝜎(X^#, X)) is complete. Conversely, if Z is a Hausdorff locally convex TVS with continuous dual space Z', then (Z, 𝜎(Z, Z')) is complete if and only if Z = (Z')^# (i.e. the map Z → (Z')^# defined by sending z ∈ Z to the evaluation map at z (i.e. z' ↦ z'(z)) is a bijection).

In particular, if Y is a vector subspace of X^# such that Y separates points of X, then (Y, 𝜎(Y, X)) is complete if and only if Y = X^#.

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 ↦ b(•, 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 ⟨X, Z⟩ (where ⟨x, x'⟩ := x'(x) is the natural evaluation map). In particular, in this situation we can assume without loss of generality that Y is a vector subspace of X's algebraic dual and b is the evaluation map.

Convention: Often, whenever y ↦ b(•, y) is injective (especially when (X, Y, b) forms a dual pair) then we will use the common practice of assuming 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 we may 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.[3]

Algebraic adjoint

In the spacial case where the dualities are the canonical dualities ⟨X, X^#⟩ and ⟨W, W^#⟩, the transpose of a linear map F : X → W is always well-defined. This transpose is called the algebraic adjoint of F and it will be denoted by F^#; that is, F^# = ^tF : W^# → X^#. In this case, for all w' ∈ W^#, F^#(w') := w' ∘ F[2][14] where the defining condition for F^#(w') is:

⟨x, F^#(w')⟩ = ⟨F(x), w'⟩     for all x ∈ X,

or equivalently, F^#(w')(x) = w'(F(x)) for all x ∈ X.

Examples

If X = Y = 𝕂ⁿ for some integer n, ℰ = {e₁, ..., e₁} is a basis for X with dual basis ℰ' = {e₁', ..., e₁'​} , F : 𝕂ⁿ → 𝕂ⁿ is a linear operator, and the matrix representation of F with respect to ℰ is M := (f_i,j), then the transpose of M is the matrix representation with respect to ℰ' of F^#.

Weak continuity and openness

Theorem[2] — Suppose that ⟨X, Y⟩ and ⟨W, Z⟩ are canonical pairings (so Y ⊆ X^# and Z ⊆ W^#) that are dual systems and let F : X → W be a linear map. Then F : X → W is weakly continuous if and only if it satisfies any of the following equivalent conditions:

1. F : (X, 𝜎(X, Y)) → (W, 𝜎(W, Z)) is continuous;
2. F^#(Z) ⊆ Y
3. the transpose of F, ^tF : Z → Y, with respect to ⟨X, Y⟩ and ⟨W, Z⟩ is well-defined.

If F is weakly continuous then ^tF : (Z, 𝜎(Z, W)) → (Y, 𝜎(Y, X)) will be continuous and furthermore, ^{t t}F = F.[14]

Definition: A map g : A → B between topological spaces is relatively open if g : A → Im g is an open mapping, where Im g is the range of g.[2]

Theorem[2] — Suppose that ⟨X, Y⟩ and ⟨W, Z⟩ are dual systems and F : X → W is a weakly continuous linear map. Then the following are equivalent:

1. F : (X, 𝜎(X, Y)) → (W, 𝜎(W, Z)) is relatively open;
2. The range of ^tF is 𝜎(Y, X)-closed in Y;
3. Im ^tF = (ker F)^⟂

Furthermore,

• F : X → W is injective (resp. bijective) if and only if ^tF is surjective (resp. bijective);
• F : X → W is surjective if and only if ^tF : (Z, 𝜎(Z, W)) → (Y, 𝜎(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.

Theorem[2] — Suppose that F : X → Y is a linear map between two Hausdorff locally convex TVSs.

• If F is continuous then it is weakly continuous and ^tF 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 ^tF : Y' → 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|| = ||^tF||.
• If F is continuous then F : X → Y is relatively open if and only if F is weakly relatively open (i.e. F : (X, 𝜎(X, X')) → (Y, 𝜎(Y, Y')) is relatively open) and every equicontinuous subsets of Im ^tF = ^tF(Y') is the image of some equicontinuous subsets of Y'.
• If F is continuous injection then F : X → Y is a TVS-embedding (or equivalently, a topological embedding) if and only if every equicontinuous subsets of X' is the image of some equicontinuous subsets of Y'.

Metrizability and separability

Theorem[2] — Let X be a locally convex space with continuous dual space X' and let K ⊆ X'.

1. If K is equicontinuous or 𝜎(X', X)-compact, and if D ⊆ X' is such that span D is dense in X, then the subspace topology that K inherits from (X', 𝜎(X', D)) is identical to the subspace topology that K inherits from (X', 𝜎(X', X)).
2. If X is separable and K is equicontinuous then K, when endowed with the subspace topology induced by (X', 𝜎(X', X)), is metrizable.
3. If X is separable and metrizable, then (X', 𝜎(X', X)) is separable.
4. 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 (X', 𝜎(X', X)).
5. 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

Main article: Polar topology

Starting with only the weak topology, we may obtain a range of locally convex topologies by using polar sets. 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 𝕂 and 𝒢 will be a non-empty collection of 𝜎(X, Y, b)-bounded subsets of X.

Polar topologies

Main article: Polar topology

The polar topology on Y determined by 𝒢 (and b) or the 𝒢-topology on Y is the unique topological vector space (TVS) topology on Y for which

${\displaystyle \left\{rG^{\circ }:G\in {\mathcal {G}},r>0\right\}}$

forms a subbasis of neighborhoods at the origin.[2] When Y is endowed with this 𝒢-topology then it is denoted by Y_𝒢. Observe that every polar topology is necessarily locally convex.[2] When 𝒢 is a directed set with respect to subset inclusion (i.e. if for all G, H ∈ 𝒢 there exists some K ∈ 𝒢 such that G ∪ H ⊆ K) 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 𝛥(Y, X, b) denotes a polar topology on Y then Y endowed with this topology will be denoted by Y_{𝛥(Y, X, b)}, Y_{𝛥(Y, X)} or simply Y_𝛥 (e.g. for 𝜎(X, Y, b) we'd have 𝛥 = 𝜎 so that Y_{σ(Y, X, b)}, Y_{σ(Y, X)} and Y_σ all denote Y with endowed with 𝜎(X, Y, b)).

${\displaystyle {\mathcal {G}}\subseteq {\mathcal {P}}\left(X\right)}$ ("topology of uniform convergence on ...")	Notation	Name ("topology of...")	Alternative name
finite subsets of X (or 𝜎(X, Y, b)-closed disked hulls of finite subsets of X)	𝜎(X, Y, b) s(X, Y, b)	pointwise/simple convergence	weak/weak* topology
𝜎(X, Y, b)-compact disks	τ(X, Y, b)		Mackey topology
𝜎(X, Y, b)-compact convex subsets	γ(X, Y, b)	compact convex convergence
𝜎(X, Y, b)-compact subsets (or balanced 𝜎(X, Y, b)-compact subsets)	c(X, Y, b)	compact convergence
𝜎(X, Y, b)-bounded subsets	b(X, Y, b) β(X, Y, b)	bounded convergence	strong topology Strongest polar topology

Definitions involving polar topologies

Continuity

Definition:[2] We say that the linear map F : X → W is Mackey continuous (with respect to (X, Y, b) and (W, Z, c)) if F : (X, τ(X, Y, b)) → (W, τ(W, Z, c)) is continuous.

Definition:[2] We say that the linear map F : X → W is strongly continuous (with respect to (X, Y, b) and (W, Z, c)) if F : (X, β(X, Y, b)) → (W, β(W, Z, c)) is continuous.

Bounded subsets

Definition: We call a subset of X weakly bounded (resp. Mackey bounded, strongly bounded) if it is bounded in (X, 𝜎(X, Y, b)) (resp. bounded in (X, τ(X, Y, b)), bounded in (X, β(X, Y, b))).

Topologies compatible with a pair

If (X, Y, b) is a pairing over 𝕂 and 𝒯 is a vector topology on X then we say that 𝒯 is a topology of the pair(ing) and that it is compatible (or consistent) with the pair(ing) (X, Y, b) if it is locally convex and if the continuous dual space of (X, 𝒯) = b(•, Y).[15] If X distinguishes points of Y then by identifying Y as a vector subspace of X's algebraic dual, the defining condition becomes: (X, 𝒯)' = Y.[2] Note that some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff,[3][16] which it would have to be if Y distinguishes the points of X (which these authors assume).

The weak topology 𝜎(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. Note that if N is a normed space that is not reflexive then the usual norm topology on its continuous dual space is not compatible with the duality (N', N).[2]

Mackey-Arens theorem

Main articles: Mackey topology and Mackey space

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

Mackey-Arens theorem I[2] — Let (X, Y, b) will be a pairing such that X distinguishes the points of Y and let 𝒯 be a locally convex topology on X (not necessarily Hausdorff). Then 𝒯 is compatible with the pairing (X, Y, b) if and only if 𝒯 is a polar topology determined by some collection 𝒢 of 𝜎(Y, X, b)-compact disks that cover[17] Y.

It follows that the Mackey topology 𝜏(X, Y, b), which recall is the polar topology generated by all 𝜎(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-Arens theorem II[2] — Let (X, Y, b) will be a pairing such that X distinguishes the points of Y and let 𝒯 be a locally convex topology on X. Then 𝒯 is compatible with the pairing if and only if 𝜎(X, Y, b) ⊆ 𝒯 ⊆ 𝜏(X, Y, b).

Mackey's theorem, barrels, and closed convex sets

If X is a TVS (over ℝ or ℂ) then a half-space is a set of the form { x ∈ X : f(x) ≤ r} for some real r and some continuous real linear functional f on X.

Theorem[18] — If X is a locally convex space (over ℝ or ℂ) and if C is a non-empty closed and convex subset of X, then C is equal to the intersection of all closed half spaces containing it.

Note that 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 𝒯 and ℒ are any locally convex topologies on X with the same continuous dual spaces, then a convex subset of X is closed in the 𝒯 topology if and only if it is closed in the ℒ topology. This implies that the 𝒯-closure of any convex subset of X is equal to its ℒ-closure and that for any 𝒯-closed disk A in X, A = A°°.[2] In particular, if B is a subset of X then B is a barrel in (X, 𝒯) if and only if it is a barrel in (X, ℒ).[2]

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

Theorem[2] — Let (X, Y, b) will be a pairing such that X distinguishes the points of Y and let 𝒯 be a topology of the pair. Then a subset of X is a barrel in X if and only if it is equal to the polar of some 𝜎(Y, X, b)-bounded subset of Y.

Theorem[2][19] — Let X be a TVS.

1. 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 rB contains that set).
2. 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.

Mackey's theorem[19][2] — Suppose that (X, 𝒯) is a Hausdorff locally convex space with continuous dual space X' and consider the canonical duality ⟨X, X'⟩. If ℒ is any topology on X that is compatible with the duality ⟨X, X'⟩ on X then the bounded subsets of (X, 𝒯) are the same as the bounded subsets of (X, ℒ).

Examples

Space of finite sequences

Let X denote the space of all sequences of scalars r_• = (r_i)^∞
_i=1 such that r_i = 0 for all sufficiently large i. Let Y = X and define a bilinear map b : X × X → 𝕂 by

b(r_•, s_•) := Σ r_is_i.

Then 𝜎(X, X, b) = τ(X, X, b).[2] Moreover, a subset T ⊆ X is 𝜎(X, X, b)-bounded (resp. β(X, X, b)-bounded) if and only if there exists a sequence m_• = (m_i)^∞
_i=1 of positive real numbers such that |t_i| ≤ m_i for all t_• = (t_i)^∞
_i=1 ∈ T and all indices i (resp. and m_• ∈ X).[2] It follows that there are weakly bounded (i.e. 𝜎(X, X, b)-bounded) subsets of X that are not strongly bounded (i.e. not β(X, X, b)-bounded).

See also

• Dual space
• Dual topology
• Duality (mathematics)
• Pairing
• Polar set
• Polar topology
• Reductive dual pair
• Strong dual
• Weak topology

References

1. Schaefer 1999, p. 122. sfn error: no target: CITEREFSchaefer1999 (help)
2. Narici 2011, pp. 225-273. sfn error: no target: CITEREFNarici2011 (help)
3. Schaefer 1999, pp. 122–128. sfn error: no target: CITEREFSchaefer1999 (help)
4. A subset S of X is called total if for all y ∈ Y, b(s, y) = 0 for all s ∈ S implies y = 0.
5. Treves 2006, p. 195. sfn error: no target: CITEREFTreves2006 (help)
6. Schaefer 1999, pp. 123–128. sfn error: no target: CITEREFSchaefer1999 (help)
7. The weak topology on Y is the weakest TVS topology on Y making all maps b(x, •) : Y → 𝕂 continuous, as x ranges over R. We also use the dual notation of Y_{𝜎(Y, R, b)}, Y_{𝜎(Y, R)}, or simply Y_𝜎 to denote Y endowed with the weak topology 𝜎(Y, R, b). If we do not indicate what the subset R is, then by the weak topology on X we mean the weak topology on X induced by Y.
8. Narici 2011, pp. 260-264. sfn error: no target: CITEREFNarici2011 (help)
9. Narici 2011, pp. 251-253. sfn error: no target: CITEREFNarici2011 (help)
10. If G : Z → Y is a linear map then G's transpose, ^tG : X → W, is well-defined if and only if Z distinguishes points of W and b(X, G(•)) ⊆ c(W, •). In this case, for each x ∈ X, the defining condition for ^tG(x) is: b(x, G(•)) = c(^tG(x), •).
11. If H : X → Z is a linear map then H's transpose, ^tH : W → Y, is well-defined if and only if X distinguishes points of Y and c(W, H(•)) ⊆ b(•, Y). In this case, for each w ∈ W, the defining condition for ^tH(w) is: c(w, H(•)) = b(•, ^tH(w)).
12. If H : W → Y is a linear map then H's transpose, ^tH : X → Z, is well-defined if and only if W distinguishes points of Z and b(X, H(•)) ⊆ c(•, Z). In this case, for each x ∈ X, the defining condition for ^tH(x) is: c(x, H(•)) = b(•, ^tH(x)).
13. If H : Y → W is a linear map then H's transpose, ^tH : Z → X, is well-defined if and only if Y distinguishes points of X and c(H(•), Z) ⊆ b(X, •). In this case, for each z ∈ Z, the defining condition for ^tH(z) is: c(H(•), z) = b(^tH(z), •).
14. Schaefer 1999, pp. 128–130. sfn error: no target: CITEREFSchaefer1999 (help)
15. 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.
16. Treves 2006, pp. 368–377. sfn error: no target: CITEREFTreves2006 (help)
17. Recall that a collection of subsets of a set S is said to cover S if every point of S is contained in some set belonging to the collection.
18. Narici 2011, p. 200. sfn error: no target: CITEREFNarici2011 (help)
19. Treves 2006, pp. 371–372. sfn error: no target: CITEREFTreves2006 (help)

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Schmitt, Lothar M (1992). "An Equivariant Version of the Hahn–Banach Theorem". Houston J. Of Math. 18: 429–447.
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)

External links

• Duality Theory