In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a Mackey space is a
locally convex topological vector space ''X'' such that the
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
of ''X'' coincides with the
Mackey topology τ(''X'',''X′''), the
finest topology which still preserves the
continuous dual. They are named after
George Mackey.
Examples
Examples of locally convex spaces that are Mackey spaces include:
* All
barrelled spaces and more generally all
infrabarreled spaces
** Hence in particular all
bornological spaces and
reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
s
* All
metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a Metric (mathematics), metr ...
s.
** In particular, all
Fréchet spaces, including all
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s and specifically
Hilbert spaces, are Mackey spaces.
* The product, locally convex direct sum, and the inductive limit of a family of Mackey spaces is a Mackey space.
[Schaefer (1999) p. 138]
Properties
* A locally convex space
with continuous dual
is a Mackey space if and only if each convex and
-relatively compact subset of
is equicontinuous.
* The
completion of a Mackey space is again a Mackey space.
[Schaefer (1999) p. 133]
* A separated quotient of a Mackey space is again a Mackey space.
* A Mackey space need not be
separable,
complete,
quasi-barrelled, nor
-quasi-barrelled.
See also
*
*
References
*
*
*
*
*
{{mathanalysis-stub
Topological vector spaces