Dual topology
Encyclopedia
In functional analysis
and related areas of mathematics
a dual topology is a locally convex topology on a dual pair
, two vector space
s 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.
, a dual topology on 
is a locally convex topology 
so that
That is the continuous dual of
is equal to 
up to
linear isomorphism.
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
, and the finest topology is the Mackey topology
, the topology of uniform convergence on all weakly compact subsets of
.
with 
a locally convex space and 
its continuous dual then 
is a dual topology on 
if and only if
it is a topology of uniform convergence on a family of absolutely convex and weakly compact
subsets of
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a dual topology is a locally convex topology on a dual pair
Dual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
, two vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s 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
Dual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
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, a dual topology on is a locally convex topology so thatThat is the continuous dual of
is equal to up toUp to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
linear isomorphism.
Properties
- Theorem (by MackeyGeorge MackeyGeorge Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
): Given a dual pair, the bounded setBounded set (topological vector space)In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set...
s 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 MackeyGeorge Mackey
George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
and Richard Arens
Richard Friedrich Arens
Richard Friederich Arens was an American mathematician. He was born in Germany but emigrated to the USA in 1925.Arens received his Ph.D. in 1945 from Harvard University....
, characterizes all possible dual topologies on a locally convex spaces.
The theorem shows that the coarsest dual topology is the weak topology
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
, the topology of uniform convergence on all finite subsets of
, and the finest topology is the Mackey topologyMackey topology
In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual...
, the topology of uniform convergence on all weakly compact subsets of
.Mackey–Arens theorem
Given a dual pairDual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
with a locally convex space and its continuous dual then is a dual topology on if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is a topology of uniform convergence on a family of absolutely convex and weakly compact
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
subsets of