In
mathematics, 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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, a Mackey space is a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
''X'' such that the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of ''X'' coincides with the
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. In other words the Mackey topology does not ma ...
τ(''X'',''X′''), the
finest topology which still preserves the
continuous dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
. They are named after
George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Unive ...
.
Examples
Examples of locally convex spaces that are Mackey spaces include:
* All
barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a ...
s and more generally all
infrabarreled spaces
** Hence in particular all
bornological space
In 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 linear maps, in the same way that ...
s and
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an i ...
s
* All
metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
s.
** In particular, all
Fréchet spaces, including all
Banach spaces and specifically
Hilbert spaces
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
, 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
*
*
*
*
*
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Topological vector spaces