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Dual system

In mathematics, particularly in functional analysis, a dual system (also called a dual pair or duality) over a field \mathbb{K} is a triple (X, Y, b), where X and Y are vector spaces over \mathbb{K}, and b: X \times Y \to \mathbb{K} is a non-degenerate bilinear map. The non-degeneracy means that Y separates points in X (for every nonzero x \in X, there exists y \in Y with b(x, y) \neq 0) and vice versa. This structure is fundamental for defining weak topologies, polar sets, and duality theories on vector spaces, with applications in quantum mechanics and Hilbert spaces.

Fundamentals

Definition and notation

In functional analysis, a dual system, also referred to as a dual pair, is defined as an ordered pair (X, Y) of vector spaces over the same scalar field \mathbb{K}, equipped with a bilinear map \langle \cdot, \cdot \rangle: X \times Y \to \mathbb{K} that separates points of X and Y. This separation property ensures that the pairing is non-degenerate, meaning that if \langle x, y \rangle = 0 for all x \in X, then y = 0, and conversely, if \langle x, y \rangle = 0 for all y \in Y, then x = 0.^[1] The annihilator of a subset A \subset X is denoted A^\perp = \{ y \in Y \mid \langle a, y \rangle = 0 \ \forall \, a \in A \}, and a subset A \subset X is called total if A^\perp = \{0\}. Similarly, for a subset B \subset Y, the annihilator is B^\perp = \{ x \in X \mid \langle x, b \rangle = 0 \ \forall \, b \in B \}, and B is total if B^\perp = \{0\}. In particular, for a non-degenerate pairing, the annihilator of the entire space satisfies \mathrm{Ann}(X) = \{ y \in Y \mid \langle x, y \rangle = 0 \ \forall \, x \in X \} = \{0\}, and analogously \mathrm{Ann}(Y) = \{0\}.^[2] Standard notation employs \langle x, y \rangle to denote the value of the bilinear form at elements x \in X and y \in Y. The dual system (X, Y, \langle \cdot, \cdot \rangle) induces a transpose pairing on (Y, X) defined by \langle y, x \rangle^t = \langle x, y \rangle, yielding an equivalent dual system under this identification.^[2]

Pairings and dual pairings

In functional analysis, a pairing on two vector spaces X and Y over a field K is defined as a bilinear form \langle \cdot, \cdot \rangle: X \times Y \to K, which is linear in each argument separately.^[3] Such a form satisfies \langle \lambda x + x', y \rangle = \lambda \langle x, y \rangle + \langle x', y \rangle for \lambda \in K, x, x' \in X, y \in Y, and analogously for the second argument.^[4] Continuity of the pairing is not assumed in this algebraic setting.^[3] The pairing is non-degenerate if X and Y are total subsets with respect to each other, meaning that for every x \in X \setminus \{0\}, there exists y \in Y such that \langle x, y \rangle \neq 0, and conversely, for every y \in Y \setminus \{0\}, there exists x \in X such that \langle x, y \rangle \neq 0.^[4] This condition ensures that the pairing separates points in each space.^[3] A key structure induced by the pairing is the linear map \phi: Y \to X^*, where X^* denotes the algebraic dual of X (the space of all linear functionals X \to K), defined by

\phi(y)(x) = \langle x, y \rangle

for all x \in X, y \in Y.^[4] The non-degeneracy condition on the Y-side (i.e., \langle x, y \rangle = 0 for all x \in X implies y = 0) is equivalent to \phi being injective.^[3] The full non-degeneracy further requires that \phi(Y) separates points on X, meaning that for every x \in X \setminus \{0\}, there exists y \in Y such that \phi(y)(x) \neq 0.^[4] If \phi is also surjective, then it is an isomorphism, fully identifying Y with X^*.^[3] In this case, the pairing is often called a duality pairing, where Y is identified with the algebraic dual X^* via \phi, and the bilinear form corresponds to the natural evaluation \langle x, y \rangle = y(x) for y \in X^*. More generally, the non-degeneracy ensures that Y can be identified with its image \phi(Y), a subspace of X^* that separates points on X; the injectivity of \phi ensures uniqueness of the representing elements. In finite-dimensional cases, non-degeneracy alone implies \phi is an isomorphism.^[3]

Orthogonality and polar sets

In a dual system consisting of vector spaces X and Y equipped with a bilinear pairing \langle \cdot, \cdot \rangle: X \times Y \to \mathbb{K} (where \mathbb{K} = \mathbb{R} or \mathbb{C}), two elements x \in X and y \in Y are said to be orthogonal, denoted x \perp y, if \langle x, y \rangle = 0.^[5] This notion extends naturally to subsets: for a subset A \subset X, the orthogonal (or annihilator) of A in Y is the set A^\perp = \{ y \in Y \mid \langle a, y \rangle = 0 \ \forall \, a \in A \}.^[5] Symmetrically, for B \subset Y, the orthogonal of B in X is B^\perp = \{ x \in X \mid \langle x, b \rangle = 0 \ \forall \, b \in B \}.^[5] These definitions arise directly from the pairing and preserve the symmetry of the dual system, as the roles of X and Y can be interchanged. The orthogonal A^\perp is always a linear subspace of Y, and if the pairing is non-degenerate, then the double orthogonal recovers the linear span: (A^\perp)^\perp = \operatorname{span} A.^[5] This result highlights the duality between subspaces of X and their orthogonals in Y, providing an algebraic tool for decomposing spaces via perpendicularity relations. Polar sets generalize orthogonality to incorporate boundedness conditions without invoking topology. For a subset A \subset X, the (absolute) polar of A is defined as A^\circ = \{ y \in Y \mid |\langle x, y \rangle| \leq 1 \ \forall \, x \in A \}.^[5]^[6] This set is convex and balanced (absorbing scalar multiples up to modulus 1), contains the origin, and is symmetric in the dual pair (X, Y), as the polar of a subset of Y is analogously defined in X.^[5] A fundamental result is the algebraic bipolar theorem, which states that for any A \subset X, the bipolar (A^\circ)^\circ equals the convex balanced hull of A \cup \{0\} (i.e., the smallest convex balanced set containing A and the origin).^[5]^[6] This theorem underscores the closure properties under polarity operations and their role in recovering convex structures from dual pairings, with the symmetry ensuring the result holds when interchanging X and Y.

Examples

Canonical duality on vector spaces

In the context of algebraic dual systems, the canonical duality arises between a vector space X over a field K and its algebraic dual X^*, which consists of all linear functionals from X to K. The canonical pairing is the bilinear map \langle x, f \rangle = f(x) for x \in X and f \in X^*.^[4] This pairing is linear in each argument separately and serves as the fundamental bilinear form associating elements of X with their evaluations under functionals in X^*.^[4] The pairing is non-degenerate, meaning that if \langle x, f \rangle = 0 for all f \in X^*, then x = 0, and conversely, if \langle x, f \rangle = 0 for all x \in X, then f = 0.^[4] This property ensures that the spaces X and X^* separate points from each other effectively. The natural embedding \iota: X \to (X^*)^* is defined by \iota(x)(f) = f(x) for x \in X and f \in X^*, where (X^*)^* is the algebraic dual of X^*, or bidual of X. This map \iota is injective due to the non-degeneracy of the pairing, embedding X algebraically into its bidual as the image \iota(X).^[4] The image \iota(X) is a total subset of (X^*)^* with respect to the dual pair ((X^*)^*, X^*), meaning that the only functional in X^* vanishing on all elements of \iota(X) is the zero functional. This totality follows directly from the injectivity of \iota and the non-degeneracy. While the embedding is canonical and injective for any vector space, it is an isomorphism if and only if X is finite-dimensional. In the infinite-dimensional case, algebraic isomorphisms between X and (X^*)^* exist via the axiom of choice and selection of Hamel bases, but they are not canonical and depend on the choice of basis, with dimensions differing in cardinality for infinite-dimensional spaces.^[7]

Dualities on topological vector spaces

In topological vector spaces, the concept of duality extends the algebraic canonical pairing by restricting attention to continuous linear functionals, thereby incorporating the topological structure. The continuous dual X' of a topological vector space X over the scalars \mathbb{K} = \mathbb{R} or \mathbb{C} is the subspace of the algebraic dual X^* consisting of all continuous linear functionals f: X \to \mathbb{K}.^[8] This ensures that the dual respects the topology on X, distinguishing it from the full algebraic dual, which includes all linear functionals without continuity requirements.^[9] The canonical pairing on the topological setting is defined by \langle x, f \rangle = f(x) for x \in X and f \in X', forming a bilinear form that separates points in both spaces under suitable conditions.^[8] This pairing induces the weak* topology \sigma(X', X) on X', which is the coarsest topology making all evaluation maps x \mapsto \langle x, f \rangle continuous; it arises directly from the original topology on X.^[10] In this framework, equicontinuous subsets of X'—those bounded by polars of neighborhoods in X—play a key role in ensuring compactness properties, as per the Banach-Alaoglu theorem.^[8] For an absorbing subset A \subset X, the polar A^\circ is defined as A^\circ = \{ f \in X' \mid |\langle x, f \rangle| \leq 1 \ \forall x \in A \}.^[8] This set is closed, convex, and balanced in the weak* topology, and it coincides with the polar of the closed balanced convex hull of A.^[8] Polars generate polar topologies on X, such as the weak topology \sigma(X, X'), providing a uniform way to describe topologies compatible with the duality.^[8] Reflexivity in topological vector spaces occurs when the natural embedding X \to (X')', given by x \mapsto \hat{x} where \hat{x}(f) = \langle x, f \rangle, is a topological isomorphism onto its image, typically under the weak* topology on (X')'.^[8] This topological identification strengthens the algebraic reflexivity of finite-dimensional spaces but fails in general for infinite-dimensional cases without additional assumptions like completeness or local convexity; for instance, Hilbert spaces are reflexive, while certain Banach spaces are not.^[10] The distinction highlights that algebraic bidual identification does not imply topological reflexivity.^[8]

Inner product and conjugate spaces

In a real inner product space X equipped with inner product \langle \cdot, \cdot \rangle, the Riesz representation theorem establishes a dual pairing between X and its algebraic dual X^* by identifying each continuous linear functional f \in X^* with a unique element y_f \in X such that f(x) = \langle x, y_f \rangle for all x \in X. This identification is isometric when X is complete (i.e., a Hilbert space), making X self-dual under the induced norm \|x\| = \sqrt{\langle x, x \rangle}. For complex inner product spaces, the situation requires adjustment to preserve bilinearity in the pairing. Consider the conjugate space \bar{X}, which has the same underlying vector space as X but with scalar multiplication defined by \lambda \cdot \bar{y} = \overline{\lambda} y for \lambda \in \mathbb{C} and y \in X. The dual pairing is then given by \langle x, \bar{y} \rangle = \langle x, y \rangle, where the right-hand side uses the original inner product on X. The Riesz representation theorem extends to this setting, associating each continuous linear functional f \in X^* with a unique \bar{y_f} \in \bar{X} such that f(x) = \langle x, \bar{y_f} \rangle. In the case of a complex Hilbert space H, this construction yields self-duality: H \cong H^* via the conjugate-linear isometry J: H \to H^* defined by

J(x)(y) = \langle y, x \rangle

for all y \in H, where the inner product is linear in the first argument and conjugate-linear in the second. This map is antilinear in x, reflecting the complex structure, and preserves the inner product up to conjugation. Orthonormal bases in H correspond naturally to dual bases under this identification, facilitating representations in quantum mechanics and signal processing.

Weak topology

Bounded subsets and Hausdorffness

In a dual system

\langle [X, Y](/page/X&Y) \rangle

with bilinear pairing \langle \cdot, \cdot \rangle: X \times Y \to \mathbb{K}, a subset A \subset X is said to be weakly bounded (or bounded in the weak topology \sigma(X, Y)) if, for every y \in Y,

\sup_{x \in A} |\langle x, y \rangle| < \infty.

This condition ensures that A is absorbed by every basic weak neighborhood of the origin in X. Equivalently, A is weakly bounded if and only if its polar A^\circ is absorbing in Y, meaning every element of Y belongs to \lambda A^\circ for some scalar \lambda > 0.^[3] When the dual system arises from a normed space, such as X normed with Y = X^* the continuous dual equipped with the dual norm, a subset A \subset X is weakly bounded if and only if it is bounded in the norm topology on X. The forward implication follows from the uniform boundedness principle applied to the pointwise bounded family of functionals \{ \langle \cdot, y \rangle : y \in Y \}, while the converse holds since the norm bounds imply uniform control on the pairings.^[11] The weak topology \sigma(X, Y) on X is defined as the coarsest topology making all maps x \mapsto \langle x, y \rangle continuous for y \in Y. A local basis of neighborhoods of the origin $0 \in X consists of the sets

V(y_1, \dots, y_n; \varepsilon_1, \dots, \varepsilon_n) = \{ x \in X : |\langle x, y_i \rangle| < \varepsilon_i \ \forall \, i = 1, \dots, n \},

where y_1, \dots, y_n \in Y are finitely many elements and \varepsilon_1, \dots, \varepsilon_n > 0. These finite intersections generate the topology, ensuring it is locally convex when the pairing is bilinear over \mathbb{R} or \mathbb{C}.^[11] The weak topology \sigma(X, Y) is Hausdorff if and only if the pairing is separated on X, meaning Y separates points in X: for any distinct x_1, x_2 \in X with x_1 \neq x_2, there exists y \in Y such that \langle x_1, y \rangle \neq \langle x_2, y \rangle. Under this condition, the singleton \{0\} is closed in \sigma(X, Y), as the separating property allows separation of the origin from nonzero points via subbasic neighborhoods. Without separation, the topology may fail to distinguish points, rendering it non-Hausdorff.^[11]

Orthogonals, quotients, and subspaces

In the context of a dual system (X, Y, \langle \cdot, \cdot \rangle), where X and Y are vector spaces equipped with a bilinear pairing, the orthogonal complement of a subset A \subset X is defined as A^\perp = \{ y \in Y \mid \langle a, y \rangle = 0 \ \forall \, a \in A \}.^[12] Similarly, for B \subset Y, the orthogonal is B^\perp = \{ x \in X \mid \langle x, b \rangle = 0 \ \forall \, b \in B \}.^[3] These orthogonals are subspaces and play a central role in the structure of the weak topology \sigma(X, Y) on X, where basic open sets are defined by finite subsets of Y, and the analogous weak topology \sigma(Y, X) on Y.^[12] For a subspace M \subset X, the bipolar M^{\perp\perp} coincides with the weak closure of M in the topology \sigma(X, Y).^[3] In reflexive dual systems, such as those arising from reflexive Banach spaces where X = Y^* and the pairing is evaluation, a closed subspace M satisfies M^{\perp\perp} = M.^[12] This bipolar theorem ensures that orthogonals capture the closure properties essential for decomposition in weak topologies. Quotient spaces in dual systems are linked to orthogonals via an algebraic isomorphism: for a subspace A \subset X, the quotient X / A is isomorphic to the algebraic dual (A^\perp)^* of A^\perp.^[12] This is realized by the canonical map \psi: X / A \to (A^\perp)^* defined by \psi(x + A)(y) = \langle x, y \rangle for y \in A^\perp, which is well-defined since \langle a, y \rangle = 0 for a \in A.^[3] The weak topology \sigma(X / A, A^\perp) on the quotient is induced naturally from \sigma(X, Y), preserving the duality structure.^[12]

Weak representation theorem

In a dual system (X, Y, \langle \cdot, \cdot \rangle) where Y is total on X—meaning that \langle x, y \rangle = 0 for all y \in Y implies x = 0—the weak representation theorem states that every continuous linear functional f on the topological vector space (X, \sigma(X, Y)) admits a representation of the form

f(x) = \langle x, y_f \rangle

for all x \in X, where y_f \in Y is unique. This result follows from the structure of the weak topology \sigma(X, Y), which is the initial topology on X induced by the family of seminorms p_y(x) = |\langle x, y \rangle| for y \in Y. Continuity of f implies that |f(x)| \leq \sum_{i=1}^n c_i p_{y_i}(x) on some neighborhood of the origin, for finite y_1, \dots, y_n \in Y and constants c_i > 0. By linearity of f and the totality of Y, which ensures separation of points, Hahn-Banach extension arguments or direct verification on the kernel of f yield the explicit form \langle \cdot, y_f \rangle, with uniqueness arising from the orthogonality condition that \langle x, y_1 - y_2 \rangle = 0 for all x \in X implies y_1 = y_2. The theorem establishes that the continuous dual of (X, \sigma(X, Y)) is precisely Y, and equipping Y with the weak* topology \sigma(Y, X) identifies it as the topological dual space, enabling Y to fully represent the functionals on X under the weak topology. This representation underpins duality theory in topological vector spaces, facilitating the study of reflexivity and compactness in dual systems.

Transposes

Definition and properties of transposes

In the context of dual systems (X, Y) and (Z, W) equipped with duality pairings \langle \cdot, \cdot \rangle_{X,Y}: X \times Y \to \mathbb{K} and \langle \cdot, \cdot \rangle_{Z,W}: Z \times W \to \mathbb{K}, where \mathbb{K} is the scalar field, the transpose (or adjoint) of a linear map T: X \to Z is the linear map T^*: W \to Y defined by the relation

\langle T x, w \rangle_{Z,W} = \langle x, T^* w \rangle_{X,Y}

for all x \in X and w \in W.^[13]^[14] This construction identifies T^* as the unique linear map that preserves the duality pairing in the reverse direction, generalizing the dual map in the setting of algebraic dual spaces.^[14] The transpose operation exhibits key algebraic properties that reflect its contravariant nature. Specifically, for scalars \lambda \in \mathbb{[K](/page/K)} and linear maps T_1, T_2: X \to Z, the transpose satisfies (\lambda T)^* = \lambda T^* and (T_1 + T_2)^* = T_1^* + T_2^*, ensuring that T^* is linear as a map from W to Y.^[13] Furthermore, if S: Z \to U is another linear map between dual systems (Z, W) and (U, V), then the composition satisfies (S T)^* = T^* S^*, reversing the order of application.^[13]^[14] In the algebraic sense, T^* corresponds to the identification of morphisms in the dual category, where linear maps between spaces induce dual maps between their paired counterparts via the bilinear forms.^[14] Additional structural relations connect the transpose to annihilators and subspaces. The kernel of T^* coincides with the annihilator of the image of T, that is,

\ker T^* = (\operatorname{im} T)^\perp = \{ w \in W \mid \langle T x, w \rangle_{Z,W} = 0 \ \forall x \in X \}.

^[14] Moreover, the image of T^* is contained in the annihilator of the kernel of T,

\operatorname{im} T^* \subset (\ker T)^\perp = \{ y \in Y \mid \langle x, y \rangle_{X,Y} = 0 \ \forall x \in \ker T \}.

^[14] These relations highlight the orthogonal complement structure inherent in dual systems, facilitating the study of exact sequences and reflexivity without invoking topology.^[13]

Weak continuity of transposes

In dual systems \langle X, Y \rangle and \langle Z, W \rangle, a linear map T: X \to Z is continuous with respect to the weak topologies \sigma(X, Y) on X and \sigma(Z, W) on Z if and only if its algebraic transpose T^*: W \to Y, defined by \langle T x, w \rangle_{Z,W} = \langle x, T^* w \rangle_{X,Y} for all w \in W and x \in X, is continuous with respect to the weak topologies \sigma(W, Z) on W and \sigma(Y, X) on Y. This symmetry arises because the weak topology \sigma(X, Y) is the initial topology making all maps x \mapsto \langle x, y \rangle continuous for y \in Y, and the transpose relation ensures the compositions align accordingly. Regarding openness properties, if T: (X, \sigma(X, Y)) \to (Z, \sigma(Z, W)) is an open mapping, then its transpose T^* is continuous when W and Y are equipped with their respective strong dual topologies \beta(W, Z) and \beta(Y, X). This follows from the open mapping theorem adapted to weak topologies on locally convex spaces, where openness of T implies that the image under T^* of strongly closed sets in W remains weakly closed in Y, ensuring the required continuity. Weak completeness enters this discussion as a structural property ensuring stability under weak closure operations relevant to transposes. A topological vector space E is weakly complete if every closed subspace of (E, \sigma(E, E')), where E' is the topological dual, is complete in the original topology of E. Equivalently, E is complete with respect to its weak topology \sigma(E, E'). In such spaces, weakly continuous transposes preserve completeness properties of subspaces, facilitating applications like the representation of closed operators via their adjoints.

Relation to canonical duality

In the setting of canonical duality between a vector space X and its algebraic dual X^*, equipped with the natural evaluation pairing \langle x, \phi \rangle = \phi(x) for x \in X and \phi \in X^*, the transpose of a linear map T: X \to Y between vector spaces is the induced linear map T^*: Y^* \to X^* defined by (T^* \phi)(x) = \phi(T x) for all \phi \in Y^* and x \in X.^[15] This construction preserves linearity and ensures that the pairing relation holds: \langle T x, \phi \rangle = \langle x, T^* \phi \rangle for all x \in X and \phi \in Y^*.^[15] The algebraic adjoint, often denoted T^\dagger: Y^* \to X^* and coinciding with T^* in this context, is explicitly given by T^\dagger(g) = g \circ T for g \in Y^*.^[16] This adjoint map is always well-defined algebraically without requiring any topology on X or Y. The Mackey topology \tau(X, X^*) is metrizable if X^* admits a countable total subset, meaning a countable subset whose linear span is dense in the weak* topology.^[17] In this canonical framework, if the pairing separates points on Y, there is a natural algebraic embedding of Y into X^* given by y \mapsto \hat{y}, where \hat{y}(x) = \langle x, y \rangle. This embedding is a homeomorphism onto its image when Y is endowed with the weak topology \sigma(Y, X) and X^* with the weak* topology \sigma(X^*, X).^[18]

Polar topologies

Definitions and bounded subsets

In a dual system \langle X, Y \rangle consisting of two vector spaces over \mathbb{R} or \mathbb{C} equipped with a bilinear pairing \langle \cdot, \cdot \rangle : X \times Y \to \mathbb{K} that separates points, assume compatible locally convex topologies on X and Y. The polar topology \gamma(X, Y) on X is the topology of uniform convergence on equicontinuous subsets of Y, generated by the seminorms p_C(x) = \sup_{y \in C} |\langle x, y \rangle| for equicontinuous C \subset Y. A local basis of convex balanced neighborhoods of the origin in \gamma(X, Y) consists of the polars of equicontinuous subsets of Y, i.e., sets of the form C^\circ = \{ x \in X \mid \sup_{y \in C} |\langle x, y \rangle| \leq 1 \}. Finite sets are equicontinuous, so polars of finite subsets form part of the basis, but the full basis includes polars of all equicontinuous sets.^[19] The polar of a subset A \subset X is the set

A^\circ = \{ y \in Y \mid |\langle x, y \rangle| \leq 1 \ \forall \, x \in A \}.

This construction yields an absolutely convex, absorbing set in Y when A generates X algebraically, and it forms the basis for generating the topology \gamma(X, Y). A subset C \subset Y is equicontinuous if its polar C^\circ is a neighborhood of the origin in the weak topology \sigma(X, Y). In the special case where Y is the topological dual of a normed space X and equipped with its norm \|\cdot\|_Y, the polar of the closed unit ball \overline{B}_Y = \{ y \in Y \mid \|y\|_Y \leq 1 \} coincides with the closed unit ball in X, given by \overline{B}_Y^\circ = \{ x \in X \mid |\langle x, y \rangle| \leq 1 \ \forall \, y \in \overline{B}_Y \} = \overline{B}_X. However, in the general algebraic dual system without a norm on Y, the definition remains the uniform bound without reference to \|y\|.^[5] The topology \gamma(X, Y) is locally convex, as it admits a generating family of seminorms, ensuring the existence of a basis of convex neighborhoods at the origin. If X is complete with respect to a compatible coarser topology (such as the Mackey topology in barrelled spaces), then (X, \gamma(X, Y)) inherits completeness; in general settings, completeness depends on the underlying structure of the dual system.^[20] A subset B \subset X is bounded in the polar topology \gamma(X, Y) if, for every neighborhood U of the origin, there exists \lambda > 0 such that B \subset \lambda U. In polar topologies, the bounded subsets of X are precisely the polars of equicontinuous subsets of Y. By the bipolar theorem, this establishes a lattice isomorphism between the saturated family of bounded subsets of X and the equicontinuous subsets of Y via the polar map. Notably, in polar topologies, every finite-dimensional subspace of X is absorbed by the family of bounded sets, as finite-dimensional subsets of Y are equicontinuous and their polars cover such subspaces through scalar multiples.^[19]^[5]

Topologies compatible with pairings

In the context of a dual system (X, Y, \langle \cdot, \cdot \rangle), a topology \tau on X is said to be compatible with the pairing if the continuous linear functionals on (X, \tau) coincide exactly with the set \{ \langle \cdot, y \rangle \mid y \in Y \}.^[21] Similarly, a topology \sigma on Y is compatible if the continuous linear functionals on (Y, \sigma) are precisely \{ \langle x, \cdot \rangle \mid x \in X \}. These compatible topologies ensure that the dual pair structure is preserved under the topological dual operation.^[21] Compatible topologies on X and Y are necessarily locally convex and Hausdorff. Moreover, the bilinear pairing \langle \cdot, \cdot \rangle: X \times Y \to \mathbb{K} is continuous when X and Y are endowed with compatible topologies, meaning that for every neighborhood V of 0 in \mathbb{K}, there exist neighborhoods U \subset X and W \subset Y such that \langle u, w \rangle \in V for all u \in U, w \in W.^[21] This continuity property facilitates the study of convergence and boundedness in duality theory. Among compatible topologies, the Mackey topology \tau_m(X, Y) on X stands out as the strongest locally convex topology compatible with the pairing.^[21] It is generated by the family of seminorms p_K(x) = \sup_{y \in K} |\langle x, y \rangle|, where K ranges over all nonempty convex balanced subsets of Y that are compact in the weak^* topology \sigma(Y, X). The dual Mackey topology on Y is defined analogously.^[21] A locally convex topology \tau on X finer than the weak topology \sigma(X, Y) is compatible with the dual pair if and only if its neighborhoods of the origin are precisely the polars of the neighborhoods of the weak^* topology \sigma(Y, X) on Y. This characterization, often expressed as \tau = \{ V^\circ \mid V \in \mathcal{N}_{\sigma(Y,X)}(0) \}, where V^\circ denotes the polar of V, underscores the duality between topologies on X and Y.^[21]

Mackey–Arens and Mackey's theorems

The Mackey–Arens theorem characterizes the locally convex Hausdorff topologies compatible with a dual system (X, Y), where the pairing separates points. It asserts that every such topology \tau on X satisfies \sigma(X, Y) \subset \tau \subset \gamma(X, Y), where \sigma(X, Y) is the weak topology generated by the pairing and \gamma(X, Y) is the topology of uniform convergence on equicontinuous subsets of Y.^[22] Equivalently, the continuous dual of (X, \tau) coincides with Y, and \tau lies between these extremes.^[23] The proof of the Mackey–Arens theorem relies on the bipolar theorem for convex sets in the context of the dual system. For an absolutely convex set B \subset X, the bipolar B^{\circ \circ} with respect to the pairing equals the \tau-closure of the convex balanced hull of B for any compatible topology \tau. This implies that equicontinuous sets in Y (whose polars are barrels in X) determine the bounding behavior across all compatible topologies, bounding \tau between \sigma(X, Y) and \gamma(X, Y).^[22] A barrel in a topological vector space is an absorbing, convex, balanced, and closed set; such sets absorb all bounded subsets by definition. A space is barrelled if every barrel is a neighborhood of the origin.^[22] Mackey's theorem states that in a complete barrelled locally convex space X with continuous dual X', the strong topology \beta(X, X') coincides with the Mackey topology \tau(X, X'). This follows because barrelledness ensures that equicontinuous subsets of X' absorb bounded sets in a way that aligns uniform convergence on them with uniform convergence on bounded sets, generating the same topology.^[22] For a compatible topology \tau on X, the continuous functionals are those y \in Y such that \{x \in X : |\langle x, y \rangle| \leq 1\} is a \tau-neighborhood of the origin, ensuring the dual cone matches across compatible topologies.^[23]

References

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Valuable practical experience – thanks to the dual system, apprentices learn directly in the company. Secure prospects of permanent employment – many people who ...
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The German dual system of vocational training combines theory and practice, knowledge and skills, learning and working in a particularly efficient manner.
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Germany's dual system of vocational education and training combines the benefits of theoretical and practical learning.
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The dual system of vocational training comprises parallel training in a company and in a vocational school. Source: Based on: Federal Government of Germany, ...
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[PDF] TOPOLOGICAL VECTOR SPACES, DISTRIBUTIONS AND KERNELS
To the student who works his way through Part I. (Topological Vector Spaces-Spaces of Functions) and Part I1 (Duality. -Spaces of Distributions), all the ...
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[PDF] Chapter 8 The Dual Space, Duality - UPenn CIS
Definition 8.1. Given a vector space E, the vector space Hom(E,K) of linear maps from E to K is called the dual space (or dual) of E. The space Hom(E,K) is.
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[PDF] Duality Theory I - KSU Math
Definitions. Suppose a dual pairing between two vector spaces X and Y is given, defined by the maps (1) and (2) satisfying (3). For a non-empty set A⊂X, we ...
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[PDF] CONVEX ANALYSIS AND NONLINEAR OPTIMIZATION Theory and ...
∗ (Bipolar theorem) For any nonempty set K ⊂ E, prove the set cl (conv (R+K)) is the smallest closed convex cone containing K. De- duce Theorem 3.3.14 ...
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[PDF] Infinite-Dimensional Linear Algebra and Solvability of Partial ...
We shortly discuss the properties of the algebraic dual V∗ of an infinite-dimensional vector space V. Both V∗ and V∗∗ are proper vector space extensions of V.
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[PDF] Introduction to topological vector spaces
Dec 5, 2016 · (In older literature, this is called the absolute polar.) For example, for Euclidean space paired with itself the polar of the ball of radius r ...
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[PDF] Locally Convex Vector Spaces V: Linear Continuous Maps and ...
(X,K), is referred to as the (topological) T -dual space of X, and is denoted simply by (X,T)∗, or just X∗, when there is no danger of confusion. 1. Page 2. The ...
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[PDF] CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real ...
DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X∗, or (X, T )∗, of X we mean the set of all continuous linear ...
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[PDF] Linear Topological Spaces
Additional titles will be luted and announced as published. Page 3. Linear Topological. Spaces by. J. L. KELLEY. ISAAC NAMIOKA. AND ... linear topological spaces.
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[PDF] FUNCTIONAL ANALYSIS | Second Edition Walter Rudin
Rudin, Walter, (date). Functional analysis/Walter Rudin.-2nd ed. p. em. -(international series in pure and applied mathematics). Includes bibliographical ...
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[PDF] A Short Course on Duality, Adjoint Operators, Green's Functions ...
Aug 6, 2004 · The concept of duality starts with linear functionals. A linear functional is just a special kind of linear map. Definition 1.23. A linear ...
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[PDF] Notes on Linear Algebra and Functional Analysis
Jul 19, 2024 · These notes are a summary of topics in linear algebra and functional analysis that are relevant to problems in Signals, Systems, ...
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245B, Notes 6: Duality and the Hahn-Banach theorem - Terry Tao
Jan 26, 2009 · The Hahn-Banach theorem is used to resolve this, by providing a powerful means to construct continuous linear functionals as needed. Theorem 1.
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[PDF] Lecture 35: Dual vector spaces. Transpose
Aim lecture: We generalise the notion of transposes of matrices to arbitrary linear maps by introducing dual vector spaces. In most of this lecture, we allow F ...<|control11|><|separator|>
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[PDF] Notes for Functional Analysis (CIMAT, Fall 2020)
Nov 23, 2021 · For the other containment assume a linear functionals F on X which is continuous with respect to the weak topology generated by F. By continuity ...
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[PDF] Approximation Properties of Locally Convex Spaces and the ... - arXiv
Aug 8, 2011 · Mackey topology τ(E′,E). The class of Fredholm operators is independent ... metrizable simple convergence on a countable total subset in E.
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[PDF] TOPOLOGICAL VECTOR SPACES, DISTRIBUTIONS AND KERNELS
The book concentrates on what, in the author's opinion, are key notions and key results (quotient spaces, transposes, the open mapping theorem, and the theorems ...
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Polars, Bipolar Theorem, Polar Topologies | SpringerLink
Mar 7, 2020 · Such a collection M defines a 'polar topology' on E, where the corresponding neighbourhoods of zero in E are polars of the members of M .
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[PDF] Functional Analysis Princeton University MAT520 Lecture Notes
Aug 18, 2023 · These lecture notes correspond to a course given in the Fall semester of 2025 in the math department of Princeton. University. Contents. 1 Soft ...
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[PDF] Dality Theory II: Advanced Theory
Besides the weak dual topologies w# introduced in DT I, there exist many other natural topologies on dual pairs. A general recipe for constructing a large ...
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An Elementary Version of the Mackey-Arens Theorem on JSTOR
Albert Wilansky, An Elementary Version of the Mackey-Arens Theorem, The American Mathematical Monthly, Vol. 75, No. 6 (Jun. - Jul., 1968), pp. 654-655.Missing: original paper