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In
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the strong dual space of a
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(TVS) $X$ is the
continuous dual space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$X^$ 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 $b\left\left(X^, X\right\right)$ or $\beta\left\left(X^, X\right\right).$ The coarsest polar topology is called
weak topology In mathematics, weak topology is an alternative term for certain initial topology, 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 initia ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s $\R$ or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... s $\C.$

## Definition from a dual system

Let $\left(X, Y, \langle \cdot, \cdot \rangle\right)$ be a
dual pair In the field of functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction i ...
of vector spaces over the field $\mathbb$ of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s $\R$ or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... 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 spa ...
topology. Let $\mathcal$ denote the
family In , family (from la, familia) is a of people related either by (by recognized birth) or (by marriage or other relationship). The purpose of families is to maintain the well-being of its members and of society. Ideally, families would off ...
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 $\beta\left(Y, X, \langle \cdot, \cdot \rangle\right)$ on $Y,$ also denoted by $b\left(Y, X, \langle \cdot, \cdot \rangle\right)$ or simply $\beta\left(Y, X\right)$ or $b\left(Y, X\right)$ 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 spa ...
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 $\left\left(X, X^, \langle \cdot , \cdot \rangle\right\right)$ where $\left\langle x, x^ \right\rangle := x^\left(x\right).$ In the special case when $X$ is a
locally convex space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
, the on the (continuous)
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
$X^$ (that is, on the space of all continuous linear functionals $f : X \to \mathbb$) is defined as the strong topology $\beta\left\left(X^, X\right\right),$ and it coincides with the topology of uniform convergence on
bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology) In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, G ...
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) In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, G ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(TVS) over the field $\mathbb.$ Let $\mathcal$ be any fundamental system of bounded sets of $X$; that is, $\mathcal$ is a
family In , family (from la, familia) is a of people related either by (by recognized birth) or (by marriage or other relationship). The purpose of families is to maintain the well-being of its members and of society. Ideally, families would off ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s on $X^$: $\left, x^\_ := \sup_ \left, x^\left(x\right)\$ as $B$ ranges over $\mathcal.$ If $X$ is
normable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
then so is $X^_$ and $X^_$ will in fact be a
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. If $X$ is a normed space with norm $\, \cdot \,$ then $X^$ has a canonical norm (the
operator norm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
) given by $\left\, x^ \right\, := \sup_ \left, x^\left(x\right) \$; 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 $X$: where the vector space $X^$ is endowed with the strong dual topology $b\left\left(X^, X^_b\right\right).$

# 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 spa ...
TVS. * A convex
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
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 then $X$'s topology is identical to the strong dual topology $b\left\left(X, X^\right\right)$ and to the
Mackey topologyIn functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional an ...
on $X.$ * If $X$ is a metrizable locally convex space, then the strong dual of $X$ is a
bornological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
if and only if it is an infrabarreled space, if and only if it is a barreled space. * If $X$ is Hausdorff locally convex TVS then $\left\left(X, b\left\left(X, X^\right\right)\right\right)$ is
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
(e.g.
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
or
LF-spaceIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
) then $X^_$ is complete. If $X$ is a barrelled space, then its topology coincides with the strong topology $\beta\left\left(X, X^\right\right)$ on $X$ and with the
Mackey topologyIn functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional an ...
on generated by the pairing $\left\left(X, X^\right\right).$

# Examples

If $X$ is a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Conversely $\left\left(X, X^\right\right).$-topology on $X$ is identical to the topology induced by the
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
on $X.$