Banach space
Normed vector space that is complete
In mathematics , more specifically in functional analysis , a Banach space (pronounced [ˈbanax] ) 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 vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
Banach spaces are named after the Polish mathematician Stefan Banach , who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly . [1] Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space ". [2] Banach spaces originally grew out of the study of function spaces by Hilbert , Fréchet , and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis , the spaces under study are often Banach spaces.
Definition
A
Banach space
is a
complete
normed space
A normed space is a pair
[note 1]
consisting of a
vector space
over a scalar field
(where
is commonly
or
) together with a distinguished
[note 2]
norm
Like all norms, this norm induces a
translation invariant
[note 3]
distance function
, called the
canonical
or
(
norm
)
induced metric
, defined for all vectors
by
[note 4]
This makes
into a
metric space
A sequence
is called
Cauchy in
or
-Cauchy
or
-Cauchy
if for every real
there exists some index
such that
whenever
and
are greater than
The normed space
is called a
Banach space
and the canonical metric
is called a
complete metric
if
is a
complete metric space
, which by definition means for every
Cauchy sequence
in
there exists some
such that
where because
this sequence's convergence to
can equivalently be expressed as:
The norm
of a normed space
is called a
complete norm
if
is a Banach space.
L-semi-inner product
For any normed space
there exists an
L-semi-inner product
on
such that
for all
; in general, there may be infinitely many L-semi-inner products that satisfy this condition. L-semi-inner products are a generalization of
inner products
, which are what fundamentally distinguish
Hilbert spaces
from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.
Characterization in terms of series
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging
series of vectors
.
A normed space
is a Banach space if and only if each
absolutely convergent
series in
converges in
[3]
Topology
The canonical metric
of a normed space
induces the usual
metric topology
on
which is referred to as the
canonical
or
norm induced
topology
.
Every normed space is automatically assumed to carry this
Hausdorff
topology, unless indicated otherwise.
With this topology, every Banach space is a
Baire space
, although there exist normed spaces that are Baire but not Banach.
[4]
The norm
is always a
continuous function
with respect to the topology that it induces.
The open and closed balls of radius
centered at a point
are, respectively, the sets
Any such ball is a
convex
and
bounded subset
of
but a
compact
ball/
neighborhood
exists if and only if
is a
finite-dimensional vector space
.
In particular, no infinite–dimensional normed space can be
locally compact
or have the
Heine–Borel property
.
If
is a vector and
is a scalar then
Using
shows that this norm-induced topology is
translation invariant
, which means that for any
and
the subset
is
open
(respectively,
closed
) in
if and only if this is true of its translation
Consequently, the norm induced topology is completely determined by any
neighbourhood basis
at the origin. Some common neighborhood bases at the origin include:
where
is a sequence in of positive real numbers that converges to
in
(such as
or
for instance).
So for example, every open subset
of
can be written as a union
indexed by some subset
where every
may be picked from the aforementioned sequence
(the open balls can be replaced with closed balls, although then the indexing set
and radii
may also need to be replaced).
Additionally,
can always be chosen to be
countable
if
is a
separable space
, which by definition means that
contains some countable
dense subset
.
Homeomorphism classes of separable Banach spaces
All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic.
Every separable infinite–dimensional
Hilbert Space
is linearly isometrically isomorphic to the separable Hilbert
sequence space
with its
usual norm
The
Anderson–Kadec theorem
states that every infinite–dimensional separable
Fréchet space
is
homeomorphic
to the
product space
of countably many copies of
(this homeomorphism need not be a
linear map
).
[5]
[6]
Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique
up to
a homeomorphism).
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including
In fact,
is even
homeomorphic
to its own
unit
sphere
which stands in sharp contrast to finite–dimensional spaces (the
Euclidean plane
is not homeomorphic to the
unit circle
, for instance).
This pattern in
homeomorphism classes
extends to generalizations of
metrizable
(
locally Euclidean
)
topological manifolds
known as
metric
Banach manifolds
, which are
metric spaces
that are around every point,
locally homeomorphic
to some open subset of a given Banach space (metric
Hilbert manifolds
and metric
Fréchet manifolds
are defined similarly).
[6]
For example, every open subset
of a Banach space
is canonically a metric Banach manifold modeled on
since the
inclusion map
is an
open
local homeomorphism
.
Using Hilbert space
microbundles
, David Henderson showed
[7]
in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or
Fréchet
) space can be
topologically embedded
as an
open
subset
of
and, consequently, also admits a unique
smooth structure
making it into a
Hilbert manifold
.
Compact and convex subsets
There is a compact subset
of
whose
convex hull
is
not
closed and thus also
not
compact (see this footnote
[note 5]
for an example).
[8]
However, like in all Banach spaces, the
closed
convex hull
of this (and every other) compact subset will be compact.
[9]
But if a normed space is not complete then it is in general
not
guaranteed that
will be compact whenever
is; an example
[note 5]
can even be found in a (non-complete)
pre-Hilbert
vector subspace of
As a topological vector space
This norm-induced topology also makes
into what is known as a
topological vector space
(TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS
is
only
a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is
not
associated with
any
particular norm or metric (both of which are "
forgotten
"). This Hausdorff TVS
is even
locally convex
because the set of all open balls centered at the origin forms a
neighbourhood basis
at the origin consisting of convex
balanced
open sets. This TVS is also
normable
, which by definition refers to any TVS whose topology is induced by some (possibly unknown)
norm
. Normable TVSs
are characterized by
being Hausdorff and having a
bounded
convex
neighborhood of the origin.
All Banach spaces are
barrelled spaces
, which means that every
barrel
is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the
Banach–Steinhaus theorem
holds.
Comparison of complete metrizable vector topologies
The
open mapping theorem
implies that if
and
are topologies on
that make both
and
into
complete metrizable TVS
(for example, Banach or
Fréchet spaces
) and if one topology is
finer or coarser
than the other then they must be equal (that is, if
or
then
).
[10]
So for example, if
and
are Banach spaces with topologies
and
and if one of these spaces has some open ball that is also an open subset of the other space (or equivalently, if one of
or
is continuous) then their topologies are identical and their
norms are equivalent
.
Completeness
Complete norms and equivalent norms
Two norms,
and
on a vector space
are said to be
equivalent
if they induce the same topology;
[11]
this happens if and only if there exist positive real numbers
such that
for all
If
and
are two equivalent norms on a vector space
then
is a Banach space if and only if
is a Banach space.
See this footnote for an example of a continuous norm on a Banach space that is
not
equivalent to that Banach space's given norm.
[note 6]
[11]
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.
[12]
Complete norms vs complete metrics
A metric
on a vector space
is induced by a norm on
if and only if
is
translation invariant
[note 3]
and
absolutely homogeneous
, which means that
for all scalars
and all
in which case the function
defines a norm on
and the canonical metric induced by
is equal to
Suppose that
is a normed space and that
is the norm topology induced on
Suppose that
is
any
metric
on
such that the topology that
induces on
is equal to
If
is
translation invariant
[note 3]
then
is a Banach space if and only if
is a complete metric space.
[13]
If
is
not
translation invariant, then it may be possible for
to be a Banach space but for
to
not
be a complete metric space
[14]
(see this footnote
[note 7]
for an example). In contrast, a theorem of Klee,
[15]
[16]
[note 8]
which also applies to all
metrizable topological vector spaces
, implies that if there exists
any
[note 9]
complete metric
on
that induces the norm topology
on
then
is a Banach space.
A
Fréchet space
is a
locally convex topological vector space
whose topology is induced by some translation-invariant complete metric.
Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the
space of real sequences
with the
product topology
).
However, the topology of every Fréchet space is induced by some
countable
family of real-valued (necessarily continuous) maps called
seminorms
, which are generalizations of
norms
.
It is even possible for a Fréchet space to have a topology that is induced by a countable family of
norms
(such norms would necessarily be continuous)
[note 10]
[17]
but to not be a Banach/
normable space
because its topology can not be defined by any
single
norm.
An example of such a space is the
Fréchet space
whose definition can be found in the article on
spaces of test functions and distributions
.
Complete norms vs complete topological vector spaces
There is another notion of completeness besides metric completeness and that is the notion of a
complete topological vector space
(TVS) or TVS-completeness, which uses the theory of
uniform spaces
.
Specifically, the notion of TVS-completeness uses a unique translation-invariant
uniformity
, called the
canonical uniformity
, that depends
only
on vector subtraction and the topology
that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology
(and even applies to TVSs that are
not
even metrizable).
Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space.
If
is a
metrizable topological vector space
(such as any norm induced topology, for example), then
is a complete TVS if and only if it is a
sequentially
complete TVS, meaning that it is enough to check that every Cauchy
sequence
in
converges in
to some point of
(that is, there is no need to consider the more general notion of arbitrary Cauchy
nets
).
If
is a topological vector space whose topology is induced by
some
(possibly unknown) norm (such spaces are called
normable
), then
is a complete topological vector space if and only if
may be assigned a
norm
that induces on
the topology
and also makes
into a Banach space.
A
Hausdorff
locally convex topological vector space
is
normable
if and only if its
strong dual space
is normable,
[18]
in which case
is a Banach space (
denotes the
strong dual space
of
whose topology is a generalization of the
dual norm
-induced topology on the
continuous dual space
; see this footnote
[note 11]
for more details).
If
is a
metrizable
locally convex TVS, then
is normable if and only if
is a
Fréchet–Urysohn space
.
[19]
This shows that in the category of
locally convex TVSs
, Banach spaces are exactly those complete spaces that are both
metrizable
and have metrizable
strong dual spaces
.
Completions
Every normed space can be isometrically embedded onto a dense vector subspace of some Banach space, where this Banach space is called a completion of the normed space. This Hausdorff completion is unique up to isometric isomorphism.
More precisely, for every normed space
there exist a Banach space
and a mapping
such that
is an
isometric mapping
and
is dense in
If
is another Banach space such that there is an isometric isomorphism from
onto a dense subset of
then
is isometrically isomorphic to
This Banach space
is the Hausdorff
completion
of the normed space
The underlying metric space for
is the same as the metric completion of
with the vector space operations extended from
to
The completion of
is sometimes denoted by
General theory
Linear operators, isomorphisms
If
and
are normed spaces over the same
ground field
the set of all
continuous
-linear maps
is denoted by
In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space
to another normed space is continuous if and only if it is
bounded
on the closed
unit ball
of
Thus, the vector space
can be given the
operator norm
For
a Banach space, the space
is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the
function space
between two Banach spaces to only the
short maps
; in that case the space
reappears as a natural
bifunctor
.
[20]
If
is a Banach space, the space
forms a unital
Banach algebra
; the multiplication operation is given by the composition of linear maps.
If
and
are normed spaces, they are
isomorphic normed spaces
if there exists a linear bijection
such that
and its inverse
are continuous. If one of the two spaces
or
is complete (or
reflexive
,
separable
, etc.) then so is the other space. Two normed spaces
and
are
isometrically isomorphic
if in addition,
is an
isometry
, that is,
for every
in
The
Banach–Mazur distance
between two isomorphic but not isometric spaces
and
gives a measure of how much the two spaces
and
differ.
Continuous and bounded linear functions and seminorms
Every
continuous linear operator
is a
bounded linear operator
and if dealing only with normed spaces then the converse is also true. That is, a
linear operator
between two normed spaces is
bounded
if and only if it is a
continuous function
. So in particular, because the scalar field (which is
or
) is a normed space, a
linear functional
on a normed space is a
bounded linear functional
if and only if it is a
continuous linear functional
. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.
If
is a
subadditive function
(such as a norm, a
sublinear function
, or real linear functional), then
[21]
is
continuous at the origin
if and only if
is
uniformly continuous
on all of
; and if in addition
then
is continuous if and only if its
absolute value
is continuous, which happens if and only if
is an open subset of
[21]
[note 12]
And very importantly for applying the
Hahn–Banach theorem
, a linear functional
is continuous if and only if this is true of its
real part
and moreover,
and
the real part
completely determines
which is why the Hahn–Banach theorem is often stated only for real linear functionals.
Also, a linear functional
on
is continuous if and only if the
seminorm
is continuous, which happens if and only if there exists a continuous seminorm
such that
; this last statement involving the linear functional
and seminorm
is encountered in many versions of the Hahn–Banach theorem.
Basic notions
The Cartesian product
of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used,
[22]
such as
which correspond (respectively) to the
coproduct
and
product
in the category of Banach spaces and short maps (discussed above).
[20]
For finite (co)products, these norms give rise to isomorphic normed spaces, and the product
(or the direct sum
) is complete if and only if the two factors are complete.
If
is a
closed
linear subspace
of a normed space
there is a natural norm on the
quotient space
The quotient
is a Banach space when
is complete.
[23]
The
quotient map
from
onto
sending
to its class
is linear, onto and has norm
except when
in which case the quotient is the null space.
The closed linear subspace
of
is said to be a
complemented subspace
of
if
is the
range
of a
surjective
bounded linear
projection
In this case, the space
is isomorphic to the direct sum of
and
the kernel of the projection
Suppose that
and
are Banach spaces and that
There exists a
canonical factorization
of
as
[23]
where the first map
is the quotient map, and the second map
sends every class
in the quotient to the image
in
This is well defined because all elements in the same class have the same image. The mapping
is a linear bijection from
onto the range
whose inverse need not be bounded.
Classical spaces
Basic examples
[24]
of Banach spaces include: the
Lp spaces
and their special cases, the
sequence spaces
that consist of scalar sequences indexed by
natural numbers
; among them, the space
of
absolutely summable
sequences and the space
of square summable sequences; the space
of sequences tending to zero and the space
of bounded sequences; the space
of continuous scalar functions on a compact Hausdorff space
equipped with the max norm,
According to the
Banach–Mazur theorem
, every Banach space is isometrically isomorphic to a subspace of some
[25]
For every separable Banach space
there is a closed subspace
of
such that
[26]
Any
Hilbert space
serves as an example of a Banach space. A Hilbert space
on
is complete for a norm of the form
where
is the inner product , linear in its first argument that satisfies the following:
For example, the space
is a Hilbert space.
The
Hardy spaces
, the
Sobolev spaces
are examples of Banach spaces that are related to
spaces and have additional structure. They are important in different branches of analysis,
Harmonic analysis
and
Partial differential equations
among others.
Banach algebras
A
Banach algebra
is a Banach space
over
or
together with a structure of
algebra over
, such that the product map
is continuous. An equivalent norm on
can be found so that
for all
Examples
-
The Banach space
with the pointwise product, is a Banach algebra.
-
The
disk algebra
consists of functions holomorphic in the open unit disk
and continuous on its closure :
Equipped with the max norm on
the disk algebra
is a closed subalgebra of
-
The
Wiener algebra
is the algebra of functions on the unit circle
with absolutely convergent Fourier series. Via the map associating a function on
to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra
where the product is the convolution of sequences.
-
For every Banach space
the space
of bounded linear operators on
with the composition of maps as product, is a Banach algebra.
-
A
C*-algebra
is a complex Banach algebra
with an antilinear involution
such that
The space
of bounded linear operators on a Hilbert space
is a fundamental example of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some
The space
of complex continuous functions on a compact Hausdorff space
is an example of commutative C*-algebra, where the involution associates to every function
its complex conjugate
Dual space
If
is a normed space and
the underlying
field
(either the
real
or the
complex numbers
), the
continuous dual space
is the space of continuous linear maps from
into
or
continuous linear functionals
.
The notation for the continuous dual is
in this article.
[27]
Since
is a Banach space (using the
absolute value
as norm), the dual
is a Banach space, for every normed space
The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem .
Hahn–Banach theorem
—
Let
be a
vector space
over the field
Let further
-
be a linear subspace ,
-
be a sublinear function and
-
be a linear functional so that
for all
In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.
[28]
An important special case is the following: for every vector
in a normed space
there exists a continuous linear functional
on
such that
When
is not equal to the
vector, the functional
must have norm one, and is called a
norming functional
for
The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane . The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane. [29]
A subset
in a Banach space
is
total
if the
linear span
of
is
dense
in
The subset
is total in
if and only if the only continuous linear functional that vanishes on
is the
functional: this equivalence follows from the Hahn–Banach theorem.
If
is the direct sum of two closed linear subspaces
and
then the dual
of
is isomorphic to the direct sum of the duals of
and
[30]
If
is a closed linear subspace in
one can associate the
orthogonal of
in the dual,
The orthogonal
is a closed linear subspace of the dual. The dual of
is isometrically isomorphic to
The dual of
is isometrically isomorphic to
[31]
The dual of a separable Banach space need not be separable, but:
When
is separable, the above criterion for totality can be used for proving the existence of a countable total subset in
Weak topologies
The
weak topology
on a Banach space
is the
coarsest topology
on
for which all elements
in the continuous dual space
are continuous.
The norm topology is therefore
finer
than the weak topology.
It follows from the Hahn–Banach separation theorem that the weak topology is
Hausdorff
, and that a norm-closed
convex subset
of a Banach space is also weakly closed.
[33]
A norm-continuous linear map between two Banach spaces
and
is also
weakly continuous
, that is, continuous from the weak topology of
to that of
[34]
If
is infinite-dimensional, there exist linear maps which are not continuous. The space
of all linear maps from
to the underlying field
(this space
is called the
algebraic dual space
, to distinguish it from
also induces a topology on
which is
finer
than the weak topology, and much less used in functional analysis.
On a dual space
there is a topology weaker than the weak topology of
called
weak* topology
.
It is the coarsest topology on
for which all evaluation maps
where
ranges over
are continuous.
Its importance comes from the
Banach–Alaoglu theorem
.
Banach–Alaoglu theorem
—
Let
be a
normed vector space
. Then the
closed
unit ball
of the dual space is
compact
in the weak* topology.
The Banach–Alaoglu theorem can be proved using
Tychonoff's theorem
about infinite products of compact Hausdorff spaces.
When
is separable, the unit ball
of the dual is a
metrizable
compact in the weak* topology.
[35]
Examples of dual spaces
The dual of
is isometrically isomorphic to
: for every bounded linear functional
on
there is a unique element
such that
The dual of
is isometrically isomorphic to
.
The dual of
Lebesgue space
is isometrically isomorphic to
when
and
For every vector
in a Hilbert space
the mapping
defines a continuous linear functional
on
The
Riesz representation theorem
states that every continuous linear functional on
is of the form
for a uniquely defined vector
in
The mapping
is an
antilinear
isometric bijection from
onto its dual
When the scalars are real, this map is an isometric isomorphism.
When
is a compact Hausdorff topological space, the dual
of
is the space of
Radon measures
in the sense of Bourbaki.
[36]
The subset
of
consisting of non-negative measures of mass 1 (
probability measures
) is a convex w*-closed subset of the unit ball of
The
extreme points
of
are the
Dirac measures
on
The set of Dirac measures on
equipped with the w*-topology, is
homeomorphic
to
Banach–Stone Theorem
—
If
and
are compact Hausdorff spaces and if
and
are isometrically isomorphic, then the topological spaces
and
are
homeomorphic
.
[37]
[38]
The result has been extended by Amir
[39]
and Cambern
[40]
to the case when the multiplicative
Banach–Mazur distance
between
and
is
The theorem is no longer true when the distance is
[41]
In the commutative
Banach algebra
the
maximal ideals
are precisely kernels of Dirac measures on
More generally, by the
Gelfand–Mazur theorem
, the maximal ideals of a unital commutative Banach algebra can be identified with its
characters
—not merely as sets but as topological spaces: the former with the
hull-kernel topology
and the latter with the w*-topology.
In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual
Theorem
—
If
is a compact Hausdorff space, then the maximal ideal space
of the Banach algebra
is
homeomorphic
to
[37]
Not every unital commutative Banach algebra is of the form
for some compact Hausdorff space
However, this statement holds if one places
in the smaller category of commutative
C*-algebras
.
Gelfand's
representation theorem
for commutative C*-algebras states that every commutative unital
C
*-algebra
is isometrically isomorphic to a
space.
[42]
The Hausdorff compact space
here is again the maximal ideal space, also called the
spectrum
of
in the C*-algebra context.
Bidual
If
is a normed space, the (continuous) dual
of the dual
is called
bidual
, or
second dual
of
For every normed space
there is a natural map,
This defines
as a continuous linear functional on
that is, an element of
The map
is a linear map from
to
As a consequence of the existence of a
norming functional
for every
this map
is isometric, thus
injective
.
For example, the dual of
is identified with
and the dual of
is identified with
the space of bounded scalar sequences.
Under these identifications,
is the inclusion map from
to
It is indeed isometric, but not onto.
If
is
surjective
, then the normed space
is called
reflexive
(see
below
).
Being the dual of a normed space, the bidual
is complete, therefore, every reflexive normed space is a Banach space.
Using the isometric embedding
it is customary to consider a normed space
as a subset of its bidual.
When
is a Banach space, it is viewed as a closed linear subspace of
If
is not reflexive, the unit ball of
is a proper subset of the unit ball of
The
Goldstine theorem
states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual.
In other words, for every
in the bidual, there exists a
net
in
so that
The net may be replaced by a weakly*-convergent sequence when the dual
is separable.
On the other hand, elements of the bidual of
that are not in
cannot be weak*-limit of
sequences
in
since
is
weakly sequentially complete
.
Banach's theorems
Here are the main general results about Banach spaces that go back to the time of Banach's book ( Banach (1932) ) and are related to the Baire category theorem . According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space ) cannot be equal to a union of countably many closed subsets with empty interiors . Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.
Banach–Steinhaus Theorem
—
Let
be a Banach space and
be a
normed vector space
. Suppose that
is a collection of continuous linear operators from
to
The uniform boundedness principle states that if for all
in
we have
then
The Banach–Steinhaus theorem is not limited to Banach spaces.
It can be extended for example to the case where
is a
Fréchet space
, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood
of
in
such that all
in
are uniformly bounded on
The Open Mapping Theorem
—
Let
and
be Banach spaces and
be a surjective continuous linear operator, then
is an open map.
Corollary — Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.
The First Isomorphism Theorem for Banach spaces
—
Suppose that
and
are Banach spaces and that
Suppose further that the range of
is closed in
Then
is isomorphic to
This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.
Corollary
—
If a Banach space
is the internal direct sum of closed subspaces
then
is isomorphic to
This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from
onto
sending
to the sum
The Closed Graph Theorem
—
Let
be a linear mapping between Banach spaces. The graph of
is closed in
if and only if
is continuous.
Reflexivity
The normed space
is called
reflexive
when the natural map
is surjective. Reflexive normed spaces are Banach spaces.
Theorem
—
If
is a reflexive Banach space, every closed subspace of
and every quotient space of
are reflexive.
This is a consequence of the Hahn–Banach theorem.
Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space
onto the Banach space
then
is reflexive.
Theorem
—
If
is a Banach space, then
is reflexive if and only if
is reflexive.
Corollary
—
Let
be a reflexive Banach space. Then
is
separable
if and only if
is separable.
Indeed, if the dual
of a Banach space
is separable, then
is separable.
If
is reflexive and separable, then the dual of
is separable, so
is separable.
Theorem
—
Suppose that
are normed spaces and that
Then
is reflexive if and only if each
is reflexive.
Hilbert spaces are reflexive. The
spaces are reflexive when
More generally,
uniformly convex spaces
are reflexive, by the
Milman–Pettis theorem
.
The spaces
are not reflexive.
In these examples of non-reflexive spaces
the bidual
is "much larger" than
Namely, under the natural isometric embedding of
into
given by the Hahn–Banach theorem, the quotient
is infinite-dimensional, and even nonseparable.
However, Robert C. James has constructed an example
[43]
of a non-reflexive space, usually called "
the James space
" and denoted by
[44]
such that the quotient
is one-dimensional.
Furthermore, this space
is isometrically isomorphic to its bidual.
Theorem
—
A Banach space
is reflexive if and only if its unit ball is
compact
in the
weak topology
.
When
is reflexive, it follows that all closed and bounded
convex subsets
of
are weakly compact.
In a Hilbert space
the weak compactness of the unit ball is very often used in the following way: every bounded sequence in
has weakly convergent subsequences.
Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain
optimization problems
.
For example, every
convex
continuous function on the unit ball
of a reflexive space attains its minimum at some point in
As a special case of the preceding result, when
is a reflexive space over
every continuous linear functional
in
attains its maximum
on the unit ball of
The following
theorem of Robert C. James
provides a converse statement.
James' Theorem — For a Banach space the following two properties are equivalent:
-
is reflexive.
-
for all
in
there exists
with
so that
The theorem can be extended to give a characterization of weakly compact convex sets.
On every non-reflexive Banach space
there exist continuous linear functionals that are not
norm-attaining
.
However, the
Bishop
–
Phelps
theorem
[45]
states that norm-attaining functionals are norm dense in the dual
of
Weak convergences of sequences
A sequence
in a Banach space
is
weakly convergent
to a vector
if
converges to
for every continuous linear functional
in the dual
The sequence
is a
weakly Cauchy sequence
if
converges to a scalar limit
for every
in
A sequence
in the dual
is
weakly* convergent
to a functional
if
converges to
for every
in
Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the
Banach–Steinhaus
theorem.
When the sequence
in
is a weakly Cauchy sequence, the limit
above defines a bounded linear functional on the dual
that is, an element
of the bidual of
and
is the limit of
in the weak*-topology of the bidual.
The Banach space
is
weakly sequentially complete
if every weakly Cauchy sequence is weakly convergent in
It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.
Theorem
[46]
—
For every measure
the space
is weakly sequentially complete.
An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the
vector.
The
unit vector basis
of
for
or of
is another example of a
weakly null sequence
, that is, a sequence that converges weakly to
For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to
[47]
The unit vector basis of
is not weakly Cauchy.
Weakly Cauchy sequences in
are weakly convergent, since
-spaces are weakly sequentially complete.
Actually, weakly convergent sequences in
are norm convergent.
[48]
This means that
satisfies
Schur's property
.
Results involving the
basis
Weakly Cauchy sequences and the
basis are the opposite cases of the dichotomy established in the following deep result of
H.
P.
Rosenthal.
[49]
Theorem
[50]
—
Let
be a bounded sequence in a Banach space. Either
has a weakly Cauchy subsequence, or it admits a subsequence
equivalent
to the standard unit vector basis of
A complement to this result is due to Odell and Rosenthal (1975).
Theorem
[51]
—
Let
be a separable Banach space. The following are equivalent:
-
The space
does not contain a closed subspace isomorphic to
-
Every element of the bidual
is the weak*-limit of a sequence
in
By the Goldstine theorem, every element of the unit ball
of
is weak*-limit of a net in the unit ball of
When
does not contain
every element of
is weak*-limit of a
sequence
in the unit ball of
[52]
When the Banach space
is separable, the unit ball of the dual
equipped with the weak*-topology, is a metrizable compact space
[35]
and every element
in the bidual
defines a bounded function on
:
This function is continuous for the compact topology of
if and only if
is actually in
considered as subset of
Assume in addition for the rest of the paragraph that
does not contain
By the preceding result of Odell and Rosenthal, the function
is the
pointwise limit
on
of a sequence
of continuous functions on
it is therefore a
first Baire class function
on
The unit ball of the bidual is a pointwise compact subset of the first Baire class on
[53]
Sequences, weak and weak* compactness
When
is separable, the unit ball of the dual is weak*-compact by the
Banach–Alaoglu theorem
and metrizable for the weak* topology,
[35]
hence every bounded sequence in the dual has weakly* convergent subsequences.
This applies to separable reflexive spaces, but more is true in this case, as stated below.
The weak topology of a Banach space
is metrizable if and only if
is finite-dimensional.
[54]
If the dual
is separable, the weak topology of the unit ball of
is metrizable.
This applies in particular to separable reflexive Banach spaces.
Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.
Eberlein–Šmulian theorem
[55]
—
A set
in a Banach space is relatively weakly compact if and only if every sequence
in
has a weakly convergent subsequence.
A Banach space
is reflexive if and only if each bounded sequence in
has a weakly convergent subsequence.
[56]
A weakly compact subset
in
is norm-compact. Indeed, every sequence in
has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of
Type and cotype
A way to classify Banach spaces is through the probabilistic notion of type and cotype , these two measure how far a Banach space is from a Hilbert space.
Schauder bases
A
Schauder basis
in a Banach space
is a sequence
of vectors in
with the property that for every vector
there exist
uniquely
defined scalars
depending on
such that
Banach spaces with a Schauder basis are necessarily separable , because the countable set of finite linear combinations with rational coefficients (say) is dense.
It follows from the Banach–Steinhaus theorem that the linear mappings
are uniformly bounded by some constant
Let
denote the coordinate functionals which assign to every
in
the coordinate
of
in the above expansion.
They are called
biorthogonal functionals
. When the basis vectors have norm
the coordinate functionals
have norm
in the dual of
Most classical separable spaces have explicit bases.
The
Haar system
is a basis for
The
trigonometric system
is a basis in
when
The
Schauder system
is a basis in the space
[57]
The question of whether the disk algebra
has a basis
[58]
remained open for more than forty years, until Bočkarev showed in 1974 that
admits a basis constructed from the
Franklin system
.
[59]
Since every vector
in a Banach space
with a basis is the limit of
with
of finite rank and uniformly bounded, the space
satisfies the
bounded approximation property
.
The first example by
Enflo
of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.
[60]
Robert C. James characterized reflexivity in Banach spaces with a basis: the space
with a Schauder basis is reflexive if and only if the basis is both
shrinking and boundedly complete
.
[61]
In this case, the biorthogonal functionals form a basis of the dual of
Tensor product
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tensor-diagramB.jpg/220px-Tensor-diagramB.jpg)
Let
and
be two
-vector spaces. The
tensor product
of
and
is a
-vector space
with a bilinear mapping
which has the following
universal property
:
-
If
is any bilinear mapping into a
-vector space
then there exists a unique linear mapping
such that
The image under
of a couple
in
is denoted by
and called a
simple tensor
.
Every element
in
is a finite sum of such simple tensors.
There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955. [62]
In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the
projective tensor product
[63]
of two Banach spaces
and
is the
completion
of the algebraic tensor product
equipped with the projective tensor norm, and similarly for the
injective tensor product
[64]
Grothendieck proved in particular that
[65]
where
is a compact Hausdorff space,
the Banach space of continuous functions from
to
and
the space of Bochner-measurable and integrable functions from
to
and where the isomorphisms are isometric.
The two isomorphisms above are the respective extensions of the map sending the tensor
to the vector-valued function
Tensor products and the approximation property
Let
be a Banach space. The tensor product
is identified isometrically with the closure in
of the set of finite rank operators.
When
has the
approximation property
, this closure coincides with the space of
compact operators
on
For every Banach space
there is a natural norm
linear map
obtained by extending the identity map of the algebraic tensor product. Grothendieck related the
approximation problem
to the question of whether this map is one-to-one when
is the dual of
Precisely, for every Banach space
the map
is one-to-one if and only if
has the approximation property.
[66]
Grothendieck conjectured that
and
must be different whenever
and
are infinite-dimensional Banach spaces.
This was disproved by
Gilles Pisier
in 1983.
[67]
Pisier constructed an infinite-dimensional Banach space
such that
and
are equal. Furthermore, just as
Enflo's
example, this space
is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space
does not have the approximation property.
[68]
Some classification results
Characterizations of Hilbert space among Banach spaces
A necessary and sufficient condition for the norm of a Banach space
to be associated to an inner product is the
parallelogram identity
:
Parallelogram identity
—
for all
It follows, for example, that the
Lebesgue space
is a Hilbert space only when
If this identity is satisfied, the associated inner product is given by the
polarization identity
. In the case of real scalars, this gives:
For complex scalars, defining the
inner product
so as to be
-linear in
antilinear
in
the polarization identity gives:
To see that the parallelogram law is sufficient, one observes in the real case that
is symmetric, and in the complex case, that it satisfies the
Hermitian symmetry
property and
The parallelogram law implies that
is additive in
It follows that it is linear over the rationals, thus linear by continuity.
Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available.
The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant
: Kwapień proved that if
for every integer
and all families of vectors
then the Banach space
is isomorphic to a Hilbert space.
[69]
Here,
denotes the average over the
possible choices of signs
In the same article, Kwapień proved that the validity of a Banach-valued
Parseval's theorem
for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.
Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space.
[70]
The proof rests upon
Dvoretzky's theorem
about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer
any finite-dimensional normed space, with dimension sufficiently large compared to
contains subspaces nearly isometric to the
-dimensional Euclidean space.
The next result gives the solution of the so-called
homogeneous space problem
. An infinite-dimensional Banach space
is said to be
homogeneous
if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to
is homogeneous, and Banach asked for the converse.
[71]
Theorem [72] — A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space.
An infinite-dimensional Banach space is
hereditarily indecomposable
when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces.
The
Gowers
dichotomy theorem
[72]
asserts that every infinite-dimensional Banach space
contains, either a subspace
with
unconditional basis
, or a hereditarily indecomposable subspace
and in particular,
is not isomorphic to its closed hyperplanes.
[73]
If
is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and
Tomczak–Jaegermann
, for spaces with an unconditional basis,
[74]
that
is isomorphic to
Metric classification
If
is an
isometry
from the Banach space
onto the Banach space
(where both
and
are vector spaces over
), then the
Mazur–Ulam theorem
states that
must be an affine transformation.
In particular, if
this is
maps the zero of
to the zero of
then
must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.
Topological classification
Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.
Anderson–Kadec theorem (1965–66) proves [75] that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved [76] that any two Banach spaces are homeomorphic if and only if they have the same density character , the minimum cardinality of a dense subset.
Spaces of continuous functions
When two compact Hausdorff spaces
and
are
homeomorphic
, the Banach spaces
and
are isometric. Conversely, when
is not homeomorphic to
the (multiplicative) Banach–Mazur distance between
and
must be greater than or equal to
see above the
results by Amir and Cambern
.
Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:
[77]
Theorem
[78]
—
Let
be an uncountable compact metric space. Then
is isomorphic to
The situation is different for
countably infinite
compact Hausdorff spaces.
Every countably infinite compact
is homeomorphic to some closed interval of
ordinal numbers
equipped with the
order topology
, where
is a countably infinite ordinal.
[79]
The Banach space
is then isometric to
C
(⟨1,
α
⟩)
. When
are two countably infinite ordinals, and assuming
the spaces
C
(⟨1,
α
⟩)
and
C
(⟨1,
β
⟩)
are isomorphic if and only if
β
<
α
ω
.
[80]
For example, the Banach spaces
are mutually non-isomorphic.
Examples
Glossary of symbols for the table below:
-
denotes the field of real numbers
or complex numbers
-
is a compact Hausdorff space .
-
are real numbers with
that are Hölder conjugates , meaning that they satisfy
and thus also
-
is a
-algebra of sets.
-
is an algebra of sets (for spaces only requiring finite additivity, such as the ba space ).
-
is a measure with variation
A positive measure is a real-valued positive set function defined on a
-algebra which is countably additive.
Classical Banach spaces | ||||||
Dual space | Reflexive | weakly sequentially complete | Norm | Notes | ||
---|---|---|---|---|---|---|
|
|
Yes | Yes |
|
|
Euclidean space |
|
|
Yes | Yes |
|
|
|
|
|
Yes | Yes |
|
|
|
|
|
Yes | Yes |
|
|
|
|
|
No | Yes |
|
|
|
|
|
No | No |
|
|
|
|
|
No | No |
|
|
|
|
|
No | No |
|
|
Isomorphic but not isometric to
|
|
|
No | Yes |
|
|
Isometrically isomorphic to
|
|
|
No | Yes |
|
|
Isometrically isomorphic to
|
|
|
No | No |
|
|
Isometrically isomorphic to
|
|
|
No | No |
|
|
Isometrically isomorphic to
|
|
|
No | No |
|
|
|
|
|
No | No |
|
|
|
|
? | No | Yes |
|
|
|
|
? | No | Yes |
|
|
A closed subspace of
|
|
? | No | Yes |
|
|
A closed subspace of
|
|
|
Yes | Yes |
|
|
|
|
|
No | Yes |
|
|
The dual is
|
|
? | No | Yes |
|
|
|
|
? | No | Yes |
|
|
|
|
|
No | Yes |
|
|
Isomorphic to the
Sobolev space
|
|
|
No | No |
|
|
Isomorphic to
|
Derivatives
Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details. The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces . Fréchet differentiability is a stronger condition than Gateaux differentiability. The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.
Generalizations
Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions
or the space of all
distributions
on
are complete but are not normed vector spaces and hence not Banach spaces.
In
Fréchet spaces
one still has a complete
metric
, while
LF-spaces
are complete
uniform
vector spaces arising as limits of Fréchet spaces.
See also
-
Space (mathematics)
– Mathematical set with some added structure
- Fréchet space – A locally convex topological vector space that is also a complete metric space
- Hardy space – Concept within complex analysis
- Hilbert space – Type of topological vector space
- L-semi-inner product – Generalization of inner products that applies to all normed spaces
-
space – Function spaces generalizing finite-dimensional p norm spaces
- Sobolev space – Vector space of functions in mathematics
- Banach lattice – Banach space with a compatible structure of a lattice Pages displaying wikidata descriptions as a fallback
- Banach disk
-
Banach manifold
– Manifold modeled on Banach spaces
- Banach bundle – vector bundle whose fibres form Banach spaces Pages displaying wikidata descriptions as a fallback
- Distortion problem
- Interpolation space
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Modulus and characteristic of convexity
- Smith space – complete compactly generated locally convex space having a universal compact set Pages displaying wikidata descriptions as a fallback
- Topological vector space – Vector space with a notion of nearness
Notes
-
↑
It is common to read
"
is a normed space" instead of the more technically correct but (usually) pedantic "
is a normed space", especially if the norm is well known (for example, such as with
spaces ) or when there is no particular need to choose any one (equivalent) norm over any other (especially in the more abstract theory of topological vector spaces ), in which case this norm (if needed) is often automatically assumed to be denoted by
However, in situations where emphasis is placed on the norm, it is common to see
written instead of
The technically correct definition of normed spaces as pairs
may also become important in the context of category theory where the distinction between the categories of normed spaces, normable spaces , metric spaces , TVSs , topological spaces , etc. is usually important.
-
↑
This means that if the norm
is replaced with a different norm
on
then
is not the same normed space as
not even if the norms are equivalent. However, equivalence of norms on a given vector space does form an equivalence relation .
-
1
2
3
A metric
on a vector space
is said to be translation invariant if
for all vectors
This happens if and only if
for all vectors
A metric that is induced by a norm is always translation invariant.
-
↑
Because
for all
it is always true that
for all
So the order of
and
in this definition does not matter.
-
1
2
Let
be the separable Hilbert space
of square-summable sequences with the usual norm
and let
be the standard orthonormal basis (that is
at the
-coordinate). The closed set
is compact (because it is sequentially compact ) but its convex hull
is not a closed set because
belongs to the closure of
in
but
(since every sequence
is a finite convex combination of elements of
and so
for all but finitely many coordinates, which is not true of
). However, like in all complete Hausdorff locally convex spaces, the closed convex hull
of this compact subset is compact. The vector subspace
is a pre-Hilbert space when endowed with the substructure that the Hilbert space
induces on it but
is not complete and
(since
). The closed convex hull of
in
(here, "closed" means with respect to
and not to
as before) is equal to
which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might fail to be compact (although it will be precompact/totally bounded ).
-
↑
Let
denote the Banach space of continuous functions with the supremum norm and let
denote the topology on
induced by
The vector space
can be identified (via the inclusion map ) as a proper dense vector subspace
of the
space
which satisfies
for all
Let
denote the restriction of the L 1 -norm to
which makes this map
a norm on
(in general, the restriction of any norm to any vector subspace will necessarily again be a norm). The normed space
is not a Banach space since its completion is the proper superset
Because
holds on
the map
is continuous. Despite this, the norm
is not equivalent to the norm
(because
is complete but
is not).
-
↑
The
normed space
is a Banach space where the absolute value is a norm on the real line
that induces the usual Euclidean topology on
Define a metric
on
by
for all
Just like
's induced metric, the metric
also induces the usual Euclidean topology on
However,
is not a complete metric because the sequence
defined by
is a
-Cauchy sequence but it does not converge to any point of
As a consequence of not converging, this
-Cauchy sequence cannot be a Cauchy sequence in
(that is, it is not a Cauchy sequence with respect to the norm
) because if it was
-Cauchy, then the fact that
is a Banach space would imply that it converges (a contradiction). Narici & Beckenstein 2011 , pp. 47–51
-
↑
The statement of the theorem is: Let
be any metric on a vector space
such that the topology
induced by
on
makes
into a topological vector space. If
is a complete metric space then
is a complete topological vector space .
-
↑
This metric
is not assumed to be translation-invariant. So in particular, this metric
does not even have to be induced by a norm.
-
↑
A norm (or
seminorm
)
on a topological vector space
is continuous if and only if the topology
that
induces on
is coarser than
(meaning,
), which happens if and only if there exists some open ball
in
(such as maybe
for example) that is open in
-
↑
denotes the continuous dual space of
When
is endowed with the strong dual space topology , also called the topology of uniform convergence on bounded subsets of
then this is indicated by writing
(sometimes, the subscript
is used instead of
). When
is a normed space with norm
then this topology is equal to the topology on
induced by the dual norm . In this way, the strong topology is a generalization of the usual dual norm-induced topology on
-
↑
The fact that
being open implies that
is continuous simplifies proving continuity because this means that it suffices to show that
is open for
and at
(where
) rather than showing this for all real
and all
References
- ↑ Bourbaki 1987 , V.87
- ↑ Narici & Beckenstein 2011 , p. 93.
- ↑ see Theorem 1.3.9, p. 20 in Megginson (1998) .
- ↑ Wilansky 2013 , p. 29.
- ↑ Bessaga & Pełczyński 1975 , p. 189
- 1 2 Anderson & Schori 1969 , p. 315.
- ↑ Henderson 1969 .
- ↑ Aliprantis & Border 2006 , p. 185.
- ↑ Trèves 2006 , p. 145.
- ↑ Trèves 2006 , pp. 166–173.
- 1 2 Conrad, Keith. "Equivalence of norms" (PDF) . kconrad.math.uconn.edu . Archived (PDF) from the original on 2022-10-09 . Retrieved September 7, 2020 .
- ↑ see Corollary 1.4.18, p. 32 in Megginson (1998) .
- ↑ Narici & Beckenstein 2011 , pp. 47–66.
- ↑ Narici & Beckenstein 2011 , pp. 47–51.
- ↑ Schaefer & Wolff 1999 , p. 35.
- ↑ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF) . Proc. Amer. Math. Soc . 3 (3): 484–487. doi : 10.1090/s0002-9939-1952-0047250-4 . Archived (PDF) from the original on 2022-10-09.
- ↑ Trèves 2006 , pp. 57–69.
- ↑ Trèves 2006 , p. 201.
- ↑ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
- 1 2 Qiaochu Yuan (June 23, 2012). "Banach spaces (and Lawvere metrics, and closed categories)" . Annoying Precision .
- 1 2 Narici & Beckenstein 2011 , pp. 192–193.
- ↑ Banach (1932 , p. 182)
- 1 2 see pp. 17–19 in Carothers (2005) .
- ↑ see Banach (1932) , pp. 11-12.
- ↑ see Banach (1932) , Th. 9 p. 185.
- ↑ see Theorem 6.1, p. 55 in Carothers (2005)
-
↑
Several books about functional analysis use the notation
for the continuous dual, for example Carothers (2005) , Lindenstrauss & Tzafriri (1977) , Megginson (1998) , Ryan (2002) , Wojtaszczyk (1991) .
- ↑ Theorem 1.9.6, p. 75 in Megginson (1998)
- ↑ see also Theorem 2.2.26, p. 179 in Megginson (1998)
- ↑ see p. 19 in Carothers (2005) .
- ↑ Theorems 1.10.16, 1.10.17 pp.94–95 in Megginson (1998)
- ↑ Theorem 1.12.11, p. 112 in Megginson (1998)
- ↑ Theorem 2.5.16, p. 216 in Megginson (1998) .
- ↑ see II.A.8, p. 29 in Wojtaszczyk (1991)
- 1 2 3 see Theorem 2.6.23, p. 231 in Megginson (1998) .
- ↑ see N. Bourbaki, (2004), "Integration I", Springer Verlag, ISBN 3-540-41129-1 .
- 1 2 Eilenberg, Samuel (1942). "Banach Space Methods in Topology". Annals of Mathematics . 43 (3): 568–579. doi : 10.2307/1968812 . JSTOR 1968812 .
-
↑
see also
Banach (1932)
, p.
170 for metrizable
and
- ↑ Amir, Dan (1965). "On isomorphisms of continuous function spaces" . Israel Journal of Mathematics . 3 (4): 205–210. doi : 10.1007/bf03008398 . S2CID 122294213 .
- ↑ Cambern, M. (1966). "A generalized Banach–Stone theorem" . Proc. Amer. Math. Soc . 17 (2): 396–400. doi : 10.1090/s0002-9939-1966-0196471-9 . And Cambern, M. (1967). "On isomorphisms with small bound" . Proc. Amer. Math. Soc . 18 (6): 1062–1066. doi : 10.1090/s0002-9939-1967-0217580-2 .
-
↑
Cohen, H. B. (1975).
"A bound-two isomorphism between
Banach spaces" . Proc. Amer. Math. Soc . 50 : 215–217. doi : 10.1090/s0002-9939-1975-0380379-5 .
- ↑ See for example Arveson, W. (1976). An Invitation to C*-Algebra . Springer-Verlag. ISBN 0-387-90176-0 .
- ↑ R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space" . Proc. Natl. Acad. Sci. U.S.A . 37 (3): 174–177. Bibcode : 1951PNAS...37..174J . doi : 10.1073/pnas.37.3.174 . PMC 1063327 . PMID 16588998 .
- ↑ see Lindenstrauss & Tzafriri (1977) , p. 25.
- ↑ bishop, See E.; Phelps, R. (1961). "A proof that every Banach space is subreflexive" . Bull. Amer. Math. Soc . 67 : 97–98. doi : 10.1090/s0002-9904-1961-10514-4 .
- ↑ see III.C.14, p. 140 in Wojtaszczyk (1991) .
- ↑ see Corollary 2, p. 11 in Diestel (1984) .
- ↑ see p. 85 in Diestel (1984) .
- ↑ Rosenthal, Haskell P (1974). "A characterization of Banach spaces containing ℓ 1 " . Proc. Natl. Acad. Sci. U.S.A . 71 (6): 2411–2413. arXiv : math.FA/9210205 . Bibcode : 1974PNAS...71.2411R . doi : 10.1073/pnas.71.6.2411 . PMC 388466 . PMID 16592162 . Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E (1975). "On sequences spanning a complex ℓ 1 space" . Proc. Amer. Math. Soc . 47 : 515–516. doi : 10.1090/s0002-9939-1975-0358308-x .
- ↑ see p. 201 in Diestel (1984) .
- ↑ Odell, Edward W.; Rosenthal, Haskell P. (1975), "A double-dual characterization of separable Banach spaces containing ℓ 1 " (PDF) , Israel Journal of Mathematics , 20 (3–4): 375–384, doi : 10.1007/bf02760341 , S2CID 122391702 , archived (PDF) from the original on 2022-10-09 .
- ↑ Odell and Rosenthal, Sublemma p. 378 and Remark p. 379.
- ↑ for more on pointwise compact subsets of the Baire class, see Bourgain, Jean ; Fremlin, D. H.; Talagrand, Michel (1978), "Pointwise Compact Sets of Baire-Measurable Functions", Am. J. Math. , 100 (4): 845–886, doi : 10.2307/2373913 , JSTOR 2373913 .
- ↑ see Proposition 2.5.14, p. 215 in Megginson (1998) .
- ↑ see for example p. 49, II.C.3 in Wojtaszczyk (1991) .
- ↑ see Corollary 2.8.9, p. 251 in Megginson (1998) .
- ↑ see Lindenstrauss & Tzafriri (1977) p. 3.
- ↑ the question appears p. 238, §3 in Banach's book, Banach (1932) .
- ↑ see S. V. Bočkarev, "Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system". (Russian) Mat. Sb. (N.S.) 95(137) (1974), 3–18, 159.
- ↑ see Enflo, P. (1973). "A counterexample to the approximation property in Banach spaces" . Acta Math . 130 : 309–317. doi : 10.1007/bf02392270 . S2CID 120530273 .
- ↑ see R.C. James, "Bases and reflexivity of Banach spaces". Ann. of Math. (2) 52, (1950). 518–527. See also Lindenstrauss & Tzafriri (1977) p. 9.
- ↑ see A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires". Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp., and A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques". Bol. Soc. Mat. São Paulo 8 1953 1–79.
- ↑ see chap. 2, p. 15 in Ryan (2002) .
- ↑ see chap. 3, p. 45 in Ryan (2002) .
- ↑ see Example. 2.19, p. 29, and pp. 49–50 in Ryan (2002) .
- ↑ see Proposition 4.6, p. 74 in Ryan (2002) .
- ↑ see Pisier, Gilles (1983), "Counterexamples to a conjecture of Grothendieck", Acta Math. 151 :181–208.
-
↑
see Szankowski, Andrzej (1981), "
does not have the approximation property", Acta Math. 147 : 89–108. Ryan claims that this result is due to Per Enflo , p. 74 in Ryan (2002) .
- ↑ see Kwapień, S. (1970), "A linear topological characterization of inner-product spaces", Studia Math. 38 :277–278.
- ↑ Lindenstrauss, Joram; Tzafriri, Lior (1971). "On the complemented subspaces problem" . Israel Journal of Mathematics . 9 (2): 263–269. doi : 10.1007/BF02771592 .
-
↑
see p.
245 in
Banach (1932)
. The homogeneity property is called "propriété
(15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec
possède la propriété (15)".
- 1 2 Gowers, W. T. (1996), "A new dichotomy for Banach spaces", Geom. Funct. Anal. 6 :1083–1093.
- ↑ see Gowers, W. T. (1994). "A solution to Banach's hyperplane problem". Bull. London Math. Soc . 26 (6): 523–530. doi : 10.1112/blms/26.6.523 .
- ↑ see Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1995). "Banach spaces without local unconditional structure" . Israel Journal of Mathematics . 89 (1–3): 205–226. arXiv : math/9306211 . doi : 10.1007/bf02808201 . S2CID 5220304 . and also Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1998). "Erratum to: Banach spaces without local unconditional structure" . Israel Journal of Mathematics . 105 : 85–92. arXiv : math/9607205 . doi : 10.1007/bf02780323 . S2CID 18565676 .
- ↑ C. Bessaga, A. Pełczyński (1975). Selected Topics in Infinite-Dimensional Topology . Panstwowe wyd. naukowe. pp. 177–230.
- ↑ H. Torunczyk (1981). Characterizing Hilbert Space Topology . Fundamenta MAthematicae. pp. 247–262.
- ↑ Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 2 :150–156.
- ↑ Milutin. See also Rosenthal, Haskell P., "The Banach spaces C(K)" in Handbook of the geometry of Banach spaces, Vol. 2, 1547–1602, North-Holland, Amsterdam, 2003.
-
↑
One can take
α
=
ω
βn
, where
is the Cantor–Bendixson rank of
and
is the finite number of points in the
-th derived set
of
See Mazurkiewicz, Stefan ; Sierpiński, Wacław (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Mathematicae 1: 17–27.
- ↑ Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions", Studia Math. 19 :53–62.
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- "Banach space" , Encyclopedia of Mathematics , EMS Press , 2001 [1994]
- Weisstein, Eric W. "Banach Space" . MathWorld .
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