In the area of mathematics known as
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 reflexive space is a
locally convex 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) for which the canonical evaluation map from
into its
bidual
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
(which is the
strong dual
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of the strong dual of
) is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of TVSs.
Since a
normable TVS is reflexive if and only if it is
semi-reflexive, every
normed space (and so in particular, every
Banach space)
is reflexive if and only if the canonical evaluation map from
into its bidual is
surjective;
in this case the normed space is necessarily also a Banach space.
In 1951,
R. C. James discovered a Banach space, now known as
James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map).
Reflexive spaces play an important role in the general theory of
locally convex TVSs and in the theory of
Banach spaces in particular.
Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
Definition
;Definition of the bidual
Suppose that
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
(which is either the real or complex numbers) whose
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 ...
,
separates points on
(that is, for any
there exists some
such that
).
Let
and
both denote the
strong dual
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of
which is the vector space
of continuous linear functionals on
endowed with the
topology of uniform convergence In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves.
The arti ...
on
bounded subsets of
;
this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).
If
is a normed space, then the strong dual of
is the continuous dual space
with its usual norm topology.
The bidual of
denoted by
is the strong dual of
; that is, it is the space
If
is a normed space, then
is the continuous dual space of the Banach space
with its usual norm topology.
;Definitions of the evaluation map and reflexive spaces
For any
let
be defined by
where
is a linear map called the evaluation map at
;
since
is necessarily continuous, it follows that
Since
separates points on
the linear map
defined by
is injective where this map is called the evaluation map or the canonical map.
Call
semi-reflexive if
is bijective (or equivalently,
surjective) and we call
reflexive if in addition
is an isomorphism of TVSs.
A
normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.
Reflexive Banach spaces
Suppose
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 ...
over the number field
or
(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 ...
s or the
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 fo ...
s), with a norm
Consider its
dual normed space that consists of all
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s
and is equipped with the
dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual ...
defined by
The dual
is a normed space (a
Banach space to be precise), and its dual normed space
is called bidual space for
The bidual consists of all continuous linear functionals
and is equipped with the norm
dual to
Each vector
generates a scalar function
by the formula:
and
is a continuous linear functional on
that is,
One obtains in this way a map
called evaluation map, that is linear. It follows from the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
that
is injective and preserves norms:
that is,
maps
isometrically onto its image
in
Furthermore, the image
is closed in
but it need not be equal to
A normed space
is called reflexive if it satisfies the following equivalent conditions:
- the evaluation map is surjective,
- the evaluation map is an isometric isomorphism of normed spaces,
- the evaluation map is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of normed spaces.
A reflexive space
is a Banach space, since
is then isometric to the Banach space
Remark
A Banach space
is reflexive if it is linearly isometric to its bidual under this canonical embedding
James' space is an example of a non-reflexive space which is linearly isometric to its
bidual
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
. Furthermore, the image of James' space under the canonical embedding
has
codimension one in its bidual.
A Banach space
is called quasi-reflexive (of order
) if the quotient
has finite dimension
Examples
# Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection
from the definition is bijective, by the
rank–nullity theorem
The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theo ...
.
# The Banach space
of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that
and are not reflexive, because
is isomorphic to the dual of
and
is isomorphic to the dual of
# All
Hilbert spaces are reflexive, as are the
Lp spaces
for
More generally: all
uniformly convex Banach spaces are reflexive according to the
Milman–Pettis theorem. The
and
spaces are not reflexive (unless they are finite dimensional, which happens for example when
is a measure on a finite set). Likewise, the Banach space
of continuous functions on