Dual topology

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.

The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology.

Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.

Definition

Given a dual pair $(X, Y, \langle , \rangle)$ , a dual topology on $X$ is a locally convex topology $\tau$ so that

(X, \tau)' \simeq Y.

Here $(X, \tau)'$ denotes the continuous dual of $(X,\tau)$ and $(X, \tau)' \simeq Y$ means that there is a linear isomorphism

\Psi : Y \to (X, \tau)',\quad y \mapsto (x \mapsto \langle x, y\rangle).

(If a locally convex topology $\tau$ on $X$ is not a dual topology, then either $\Psi$ is not surjective or it is ill-defined since the linear functional $x \mapsto \langle x, y\rangle$ is not continuous on $X$ for some $y$ .)

Properties

Theorem (by Mackey): Given a dual pair, the bounded sets under any dual topology are identical.

Under any dual topology the same sets are barrelled.

Characterization of dual topologies

The Mackey–Arens theorem, named after George Mackey and Richard Arens, characterizes all possible dual topologies on a locally convex space.

The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of $X'$ , and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of $X'$ .

Mackey–Arens theorem

Given a dual pair $(X, X')$ with $X$ a locally convex space and $X'$ its continuous dual, then $\tau$ is a dual topology on $X$ if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of $X'$

References

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Strong (polar operator) Ultrastrong Weak (operator) Ultraweak Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Banach algebras	C-algebras Spectrum (C-algebra radius) Spectral theory

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure