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 xX the map yb(x, y) is not identically 0 and for all non-zero yY, the map xb(x, y) is not identically 0. The study of dual systems is called duality theory.

According to Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the deepest and most beautiful results of the subject."[1]

Definition, notation, and conventions

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 or the complex numbers ) and a bilinear map b : X × Y → 𝕂, which is called the bilinear map associated with the pairing[2] or simply the pairing's map/bilinear form.

Notation: For all xX, let b(x, •) : Y → 𝕂 denote the linear functional on Y defined by yb(x, y) and let b(X, •) := { b(x, •) : xX}. Similarly, for all yY, let b(•, y) : X → 𝕂 be defined by xb(x, y) and let b(•, Y) := { b(•, y) : yY}.
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 xX is such that b(x, •) = 0 then x = 0; or equivalently, for all non-zero xX, the map b(x, •) : Y → 𝕂 is not identically 0 (i.e. there exists a yY such that b(x, y) ≠ 0);
  2. X separates/distinguishes points of Y: if yY is such that b(•, y) = 0 then y = 0; or equivalently, for all non-zero yY, the map b(•, y) : X → 𝕂 is not identically 0 (i.e. there exists an xX such that b(x, y) ≠ 0).

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

A subset S of Y is called total if for all xX, b(x, s) = 0 for all sS implies x = 0. A total subset of X is defined analogously (see footnote).[note 1]

The elements xX and yY are orthogonal and write xy if b(x, y) = 0. Two sets RX and SY are orthogonal and write RS if r and s are orthogonal for all rR and sS while R is said to be orthogonal to an element yY if R is orthogonal to { y } . For RX, define the orthogonal or annihilator of R to be R := { yY : Ry } := { yY : b(R, y) = { 0 } }.

Polar sets

Throughout, (X, Y, b) will be a pairing over 𝕂. The absolute polar or polar of a subset A of X is the set:[4]

.

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

.

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.[5]

If AX then the bipolar of A, denoted by A°°, is the set °(A). Similarly, if BY 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), define a new pairing (Y, X, d) where d(y, x) := b(x, y) for all xX and all yY.[2] For the purpose of explaining the following convention, the map d will be called 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 a definition (or result) for a pairing (X, Y, b) is given then this article will omit mention of the corresponding dual definition (or result) but nevertheless use it.

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 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 by switching the order of X and Y, then it is meant that definition applied to (Y, X, d) (e.g. 𝜏(Y, X, b) actually denotes the topology 𝜏(Y, X, d)).

For instance, once the weak topology on X is defined, which is denoted by 𝜎(X, Y, b), then this definition will automatically be applied to the pairing (Y, X, d) so as to obtain the definition of the weak topology on Y, where this topology will be denoted by 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, this article 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 . 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: This article will use the common practice of denoting the restriction 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 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.

Notation: The evaluation map will be denoted by x, x' := x'(x) (rather than by c) and X, N 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 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 nN 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 to X × X' defines a pairing 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, it will be assumed without comment that it's 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#).

Inner product spaces and complex conjugate spaces

A pre-Hilbert space (H, ⋅, ⋅ ) is a dual pairing if and only if H is vector space over or H has dimension 0. Here it is assumed that the sesquilinear form ⋅, ⋅ is conjugate homogeneous in its second coordinate and homogeneous in its first coordinate.

  • If (H, ⋅, ⋅ ) is a real Hilbert space then (H, H, ⋅, ⋅ ) forms a dual system.
  • If (H, ⋅, ⋅ ) is a complex Hilbert space then(H, H, ⋅, ⋅ ) forms a dual system if and only if dim H = 0. If H is non-trivial then (H, H, ⋅, ⋅ ) doesn't even form pairing since the inner product is sesquilinear rather than bilinear.[2]

Suppose that (H, ⋅, ⋅ ) is a complex pre-Hilbert space with multiplication denoted by juxtaposition or by a dot . Define the map

• ⊥ • : ℂ × HH       by       cx := c x,

where the right hand side uses the scalar multiplication of H. Let H denote the complex conjugate vector space of H, where H denotes the additive group of (H, +) (so vector addition in H is identical to vector addition in H) but with scalar multiplication in H being the map • ⊥ • rather than the scalar multiplication that H is endowed with.

The map b : H × H → ℂ defined by b (x, y) := x, y is linear in both coordinates[note 2] and so (H, H, ⋅, ⋅ ) forms a dual pairing.

Other examples

  • Suppose X = ℝ2, Y = ℝ3, and for all X and 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 := { yY : Xy } = { (0, 0, z) : z ∈ ℝ }.
  • Let 1 < p < ∞, X := Lp(μ), Y := Lq(μ) (where q is such that 1/p + 1/q = 1), and b(f, g) := . Then (X, Y, b) is a dual system.
  • Let X and Y be vector spaces over the same field 𝕂. Then the bilinear form places X × Y and in duality.[3]
  • A sequence space X and its beta dual with the bilinear map defined as for xX, y forms a dual system.

Weak topology

Suppose that (X, Y, b) is a pairing of vector spaces over 𝕂. If SY 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] The notation X𝜎(X, S, b), X𝜎(X, S), or (if no confusion could arise) simply X𝜎 is used to denote X endowed with the weak topology 𝜎(X, S, b). If it is not indicated what the subset S is, then by the weak topology on X it is meant the weak topology on X induced by Y. Similarly, if RX then 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).[note 3] Importantly, the weak topology depends entirely on the function b and the usual topology on (the topology doesn't even depend on the algebraic structures of X and Y).

Definition and Notation: If "𝜎(X, Y, b)" is attached to a topological definition (e.g. 𝜎(X, Y, b)-converges, 𝜎(X, Y, b)-bounded, cl𝜎(X, Y, b)(S), etc.) then it means that definition when the first space (i.e. X) carries the 𝜎(X, Y, b) topology. Mention of b or even X and Y may be omitted if no confusion will arise. So for instance, if a sequence 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 py : X → ℝ defined by py(x) := |b(x, y)|, as y ranges over Y.[2] If xX and is a net in X, then 𝜎(X, Y, b)-converges to x if converges to X in (X, 𝜎(X, Y, b)).[2] A net 𝜎(X, Y, b)-converges to x if and only if for all yY, b(xi, y) converges to b(x, y). If is a sequence of orthonormal vectors in Hilbert space, then converges weakly to 0 but does not norm-converge to 0 (or any other vector).[2]

If (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 yY, where |b(S, y)| := {|b(s, y)| : s ∈ S}.

Hausdorffness

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

  1. X distinguishes points of Y;
  2. The map yb(•, 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) : yY}. Furthermore,

  1. If f is a continuous linear functional on (X, 𝜎(X, Y, b)) then there exists some yY 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 := { yY : b(x, y) = 0 for all xX } .
    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) = (Cl𝜎(Y, X, b)(span S)) = S⊥⊥⊥ and this set is 𝜎(Y, X, b)-closed;[2]
  • SS⊥⊥ = Cl𝜎(X, Y, b)(span S);[2]
    • Thus if S is a 𝜎(X, Y, b)-closed vector subspace of X then S = S⊥⊥.
  • If (Si)iI is a family of 𝜎(X, Y, b)-closed vector subspaces of X then (∩iI Si) = Cl𝜎(Y, X, b)(span (∪iI S
    i
    ))
    .[2]
  • If (Si)iI is a family of subsets of X then (∪iI Si) = ∩iI 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)).[6] Furthermore, if (X, 𝜎(X, Y, b)) is a dual system then so is (M, Y/M, b|M).[6]

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.[6]

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 𝜎(X', X)-bounded if and only if HB° for some barrel B in X.[2]

The following results are important for defining polar topologies.

If (X, Y, b) is a pairing and AX, then:[2]

  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.
    • The bipolar theorem in particular "is an indispensable tool in working with dualities."[5]
  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.

If (X, Y, b) is a pairing and 𝜏 is a locally convex topology on X that is 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.[7]

Transpose of a linear map with respect to pairings

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

For all zZ, let c(F(•), z) : X → 𝕂 be the map defined by xc(F(x), z). It is said 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 yb(•, y) from Y into the algebraic dual X# is injective), and
  2. c(F(•), Z) ⊆ b(•, Y), where c(F(•), Z) := { c(F(•), z) : zZ }.

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

tF : ZY

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. For every zZ, 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 xX.

By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form ZY,[note 4] XZ,[note 5] WY,[note 6] YW,[note 7] etc. (see footnote for details).

Properties of the transpose

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

  • tF : ZY 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 tF = F.
  • Suppose (U, V, a) is a pairing over 𝕂 and E : UX is a linear map whose transpose tE : YV is well-defined. Then the transpose of FE : UW, which is t(FE) : ZV, is well-defined and t(FE) = tEtF.
  • If F : XW is a vector space isomorphism then tF : ZY is bijective, the transpose of F−1 : WX, which is t(F−1) : YZ, is well-defined, and t(F−1) = (tF)−1.[2]
  • Let SX and let S° denotes the absolute polar of A, then:[2]
    1. [F(S)]° = (tF)−1(S°);
    2. if F(S) ⊆ T for some TW, then tF(T°) ⊆ S°;
    3. if TW is such that tF(T°) ⊆ S°, then F(S) ⊆ T°°;
    4. if TW 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 : XY is a continuous linear map, then ||F|| = ||tF||.[2]

Weak continuity

A linear map F : XW 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 : XW 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 : ZY 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 tF = 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

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. There exist Banach spaces that are not weakly-complete (despite being complete).[2]

If X is a vector space then under the canonical duality, (X#, 𝜎(X#, X)) is complete.[2] 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 zZ to the evaluation map at z (i.e. z'z'(z)) is a bijection).[2]

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 yb(•, 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 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.

Convention: Often, whenever yb(•, 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.[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 : XW 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][8] where the defining condition for F#(w') is:

x, F#(w') = F(x), w'     for all xX,

or equivalently, F#(w')(x) = w'(F(x)) for all xX.

Examples

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

Weak continuity and openness

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

  1. F : (X, 𝜎(X, Y)) → (W, 𝜎(W, Z)) is continuous;
  2. F#(Z) ⊆ Y
  3. the transpose of F, tF : ZY, 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 tF = F.[8]

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

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

  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 : XW is injective (resp. bijective) if and only if tF is surjective (resp. bijective);
  • F : XW 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.

If F : XY is a linear map between two Hausdorff locally convex topological vector spaces then:[2]

  • 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 : XY 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 : XY 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

Let X be a locally convex space with continuous dual space X' and let KX'.[2]

  1. If K is equicontinuous or 𝜎(X', X)-compact, and if DX' 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

Starting with only the weak topology, the use of polar sets produces a range of locally convex topologies. Such topologies are called polar topologies. The weak topology is the weakest topology of this range.

Throughout, (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

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

forms a subbasis of neighborhoods at the origin.[2] When Y is endowed with this 𝒢-topology then it is denoted by Y𝒢. 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 GHK) 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 Y𝛥(Y, X, b), Y𝛥(Y, X) or simply Y𝛥 (e.g. for 𝜎(X, Y, b) we'd have 𝛥 = 𝜎 so that Yσ(Y, X, b), Yσ(Y, X) and Yσ all denote Y with endowed with 𝜎(X, Y, b)).

("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

A linear map F : XW 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.[2]

A linear map F : XW 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.[2]

Bounded subsets

A subset of X is 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 𝒯 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 (X, 𝒯) = b(•, Y).[note 8] 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] Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff,[3][9] which it would have to be if 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. 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

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[note 9] 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 { xX : f(x) ≤ r} for some real r and some continuous real linear functional f on X.

Theorem  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.[10]

The above theorem implies that the closed and convex subsets of a locally convex space depend entirely on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality; that is, if 𝒯 and 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 topology. This implies that the 𝒯-closure of any convex subset of X is equal to its -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.

If X is a topological vector space then:[2][11]

  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[11][2]  Suppose that (X, 𝒯) is a Hausdorff locally convex space with continuous dual space X' and consider the canonical duality X, X'. If 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 = (ri)
i=1
such that ri = 0 for all sufficiently large i. Let Y = X and define a bilinear map b : X × X → 𝕂 by

b(r, s) := ∑ risi.

Then 𝜎(X, X, b) = τ(X, X, b).[2] Moreover, a subset TX is 𝜎(X, X, b)-bounded (resp. β(X, X, b)-bounded) if and only if there exists a sequence m = (mi)
i=1
of positive real numbers such that |ti|mi for all t = (ti)
i=1
T
and all indices i (resp. and mX).[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

Notes

  1. A subset S of X is called total if for all yY, b(s, y) = 0 for all sS implies y = 0.
  2. That b is linear in its first coordinate is obvious. Suppose c is a scalar. Then b (x, cy) = b (x, c y) = x, c y = c x, y = c b (x, y), which shows that b is linear in its second coordinate.
  3. 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 Y𝜎(Y, R, b), Y𝜎(Y, R), or simply Y𝜎 to denote Y endowed with the weak topology 𝜎(Y, R, b). If it is not indicated what the subset R is, then by the weak topology on X it is meant the weak topology on X induced by Y.
  4. If G : ZY is a linear map then G's transpose, tG : XW, is well-defined if and only if Z distinguishes points of W and b(X, G(•)) ⊆ c(W, •). In this case, for each xX, the defining condition for tG(x) is: b(x, G(•)) = c(tG(x), •).
  5. If H : XZ is a linear map then H's transpose, tH : WY, is well-defined if and only if X distinguishes points of Y and c(W, H(•)) ⊆ b(•, Y). In this case, for each wW, the defining condition for tH(w) is: c(w, H(•)) = b(•, tH(w)).
  6. If H : WY is a linear map then H's transpose, tH : XZ, is well-defined if and only if W distinguishes points of Z and b(X, H(•)) ⊆ c(•, Z). In this case, for each xX, the defining condition for tH(x) is: c(x, H(•)) = b(•, tH(x)).
  7. If H : YW is a linear map then H's transpose, tH : ZX, is well-defined if and only if Y distinguishes points of X and c(H(•), Z) ⊆ b(X, •). In this case, for each zZ, the defining condition for tH(z) is: c(H(•), z) = b(tH(z), •).
  8. 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.
  9. 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.

References

  1. Schaefer & Wolff 1999, p. 122.
  2. Narici & Beckenstein 2011, pp. 225-273.
  3. Schaefer & Wolff 1999, pp. 122–128.
  4. Trèves 2006, p. 195.
  5. Schaefer & Wolff 1999, pp. 123–128.
  6. Narici & Beckenstein 2011, pp. 260-264.
  7. Narici & Beckenstein 2011, pp. 251-253.
  8. Schaefer & Wolff 1999, pp. 128–130.
  9. Trèves 2006, pp. 368–377.
  10. Narici & Beckenstein 2011, p. 200.
  11. Trèves 2006, pp. 371–372.

Bibliography

  • 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 (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.
  • 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.
  • Schmitt, Lothar M (1992). "An Equivariant Version of the Hahn–Banach Theorem". Houston J. Of Math. 18: 429–447.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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