Strong Dual
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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 space#Definition, inner product, Norm (mathematics)#Defini ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the strong dual space of a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) X is the
continuous dual space 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 cons ...
X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X, where this topology is denoted by b\left(X^, X\right) or \beta\left(X^, X\right). The coarsest polar topology is called
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, X^, has the strong dual topology, X^_b or X^_ may be written.


Strong dual topology

Throughout, all vector spaces will be assumed to be over the field \mathbb of either the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s \C.


Definition from a dual system

Let (X, Y, \langle \cdot, \cdot \rangle) be a
dual pair In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...
of vector spaces over the field \mathbb of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s \C. For any B \subseteq X and any y \in Y, define , y, _B = \sup_, \langle x, y\rangle, . Neither X nor Y has a topology so say a subset B \subseteq X is said to be if , y, _B < \infty for all y \in C. So a subset B \subseteq X is called if and only if \sup_ , \langle x, y \rangle, < \infty \quad \text y \in Y. This is equivalent to the usual notion of bounded subsets when X is given the weak topology induced by Y, which is a Hausdorff
locally convex 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 ve ...
topology. Let \mathcal denote the
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of all subsets B \subseteq X bounded by elements of Y; that is, \mathcal is the set of all subsets B \subseteq X such that for every y \in Y, , y, _B = \sup_, \langle x, y\rangle, < \infty. Then the \beta(Y, X, \langle \cdot, \cdot \rangle) on Y, also denoted by b(Y, X, \langle \cdot, \cdot \rangle) or simply \beta(Y, X) or b(Y, X) if the pairing \langle \cdot, \cdot\rangle is understood, is defined as the
locally convex 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 ve ...
topology on Y generated by the seminorms of the form , y, _B = \sup_ , \langle x, y\rangle, ,\qquad y \in Y, \qquad B \in \mathcal. The definition of the strong dual topology now proceeds as in the case of a TVS. Note that if X is a TVS whose continuous dual space separates point on X, then X is part of a canonical dual system \left(X, X^, \langle \cdot , \cdot \rangle\right) where \left\langle x, x^ \right\rangle := x^(x). In the special case when X is a
locally convex 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 vec ...
, the on the (continuous)
dual space 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 ...
X^ (that is, on the space of all continuous linear functionals f : X \to \mathbb) is defined as the strong topology \beta\left(X^, X\right), and it coincides with the topology of uniform convergence on
bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mat ...
s in X, i.e. with the topology on X^ generated by the seminorms of the form , f, _B = \sup_ , f(x), , \qquad \text f \in X^, where B runs over the family of all
bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mat ...
s in X. The space X^ with this topology is called of the space X and is denoted by X^_.


Definition on a TVS

Suppose that X is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) over the field \mathbb. Let \mathcal be any fundamental system of bounded sets of X; that is, \mathcal is a
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of bounded subsets of X such that every bounded subset of X is a subset of some B \in \mathcal; the set of all bounded subsets of X forms a fundamental system of bounded sets of X. A basis of closed neighborhoods of the origin in X^ is given by the polars: B^ := \left\ as B ranges over \mathcal). This is a locally convex topology that is given by the set of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
s on X^: \left, x^\_ := \sup_ \left, x^(x)\ as B ranges over \mathcal. If X is
normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
then so is X^_ and X^_ will in fact be a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
. If X is a normed space with norm \, \cdot \, then X^ has a canonical norm (the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introdu ...
) given by \left\, x^ \right\, := \sup_ \left, x^(x) \; the topology that this norm induces on X^ is identical to the strong dual topology.


Bidual

The bidual or second dual of a TVS X, often denoted by X^, is the strong dual of the strong dual of X: X^ \,:=\, \left(X^_b\right)^_b where the vector space X^ is endowed with the strong dual topology b\left(X^, X^_b\right).


Properties

Let X be a
locally convex 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 ve ...
TVS. * A convex
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
weakly compact subset of X^ is bounded in X^_b. * Every weakly bounded subset of X^ is strongly bounded. * If X is a
barreled 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 ...
then X's topology is identical to the strong dual topology b\left(X, X^\right) and to 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 mak ...
on X. * If X is a metrizable locally convex space, then the strong dual of X is a
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 a ...
if and only if it is an
infrabarreled space In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infra barreled) if every bounded absorbing barrel is a neighborhood of the origin. Characteri ...
, if and only if it is a
barreled 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 ...
. * If X is Hausdorff locally convex TVS then \left(X, b\left(X, X^\right)\right) is
metrizable 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) ...
if and only if there exists a countable set \mathcal of bounded subsets of X such that every bounded subset of X is contained in some element of \mathcal. * If X is locally convex, then this topology is finer than all other \mathcal-topologies on X^ when considering only \mathcal's whose sets are subsets of X. * If X is a
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 a ...
(e.g.
metrizable 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) ...
or
LF-space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct limit ...
) then X^_ is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
. If X is a
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 b ...
, then its topology coincides with the strong topology \beta\left(X, X^\right) on X and 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 mak ...
on generated by the pairing \left(X, X^\right).


Examples

If X is a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
, then its (continuous) dual space X^ with the strong topology coincides with the Banach dual space X^; that is, with the space X^ with the topology induced by the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introdu ...
. Conversely \left(X, X^\right).-topology on X is identical to the topology induced by the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
on X.


See also

* * * * * * * *


References


Bibliography

* * * * * {{DualityInLCTVSs Functional analysis Topology of function spaces