Mackey space
Encyclopedia
In mathematics
, particularly in functional analysis
, a Mackey space is a locally convex space X such that the topology
of X coincides with the Mackey topology
τ(X,X′).
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...
, particularly in functional analysis
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...
, a Mackey space is a locally convex space X such that the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of X coincides with the Mackey topology
Mackey 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...
τ(X,X′).
Properties
- Examples of Mackey spaces include bornological spaceBornological spaceIn mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and functions, in the same way that a topological space possesses the minimum amount of...
s and barrelled spaceBarrelled spaceIn functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced,...
s. - The completion of a Mackey space is again a Mackey space.
- A separated quotient of a Mackey space is again a Mackey space.