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 defi ...
, a reflexive space is 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 v ...
topological vector space (TVS) for which the canonical evaluation map from
into its
bidual (which is the
strong dual 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
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 z ...
TVS is reflexive if and only if it is
semi-reflexive In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong du ...
, every
normed 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 ...
(and so in particular, every
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 ...
)
is reflexive if and only if the canonical evaluation map from
into its bidual is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
;
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
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 v ...
TVSs and in the theory of
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 ...
s in particular.
Hilbert space
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 natural ...
s 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 (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 of
which is the vector space
of continuous linear functionals on
endowed with the
topology of uniform convergence 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 In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong du ...
if
is bijective (or equivalently,
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
) and we call
reflexive if in addition
is an isomorphism of TVSs.
A
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 z ...
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 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 g ...
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 du ...
defined by
The dual
is a normed space (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 ...
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 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
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
,
- 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. Furthermore, the image of James' space under the canonical embedding
has
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
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 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 space
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 natural ...
s are reflexive, as are the
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
s
for
More generally: all
uniformly convex Banach spaces are reflexive according to the
Milman–Pettis theorem
In mathematics, the Milman–Pettis theorem states that every uniformly convex 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 vec ...
. 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