Dual topology

From Wikipedia, the free encyclopedia

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 spaces.

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 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 $X'$

v t e Functional analysis

Local Notions on Sets	Absolutely convex Absorbing Balanced Bounded Cones (Linear, Convex cone) Convex Radial Star-shaped Symmetric

Types of TVSs	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (Tame) Hilbert LF Locally convex Mackey Montel Nuclear Normed (Norm) Reflexive Riesz Smith Stereotype Topological tensor product (of Hilbert spaces) Webbed

Spaces of Maps	Dual space Dual topology Operator topologies Strong Topology (Polar, Operator) Topology of uniform convergence Weak topology (operator) Ultrastrong topology Ultraweak topology

Types of 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

Operations on Sets	Algebraic interior (core) Interior Minkowski addition Polar

Banach Algebras	C-algebras Spectrum (of a C-algebra, Spectral radius) Spectral theory

Important 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 Derivatives (Fréchet derivative, Gâteaux derivative, Functional, Holomorphy) Differentiation in Fréchet spaces Integrals (Bochner, Dunford, Gelfand–Pettis, Regulated, Weak) Functional calculus Inverse function theorem (Nash–Moser theorem) Vector measure

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.