In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a Banach space (pronounced ) is a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
normed vector space. Thus, a Banach space is a vector space with a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
that allows the computation of
vector length
In mathematics, a norm is a function (mathematics), function from a real number, real or complex number, complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it ...
and distance between vectors and is complete in the sense that a
Cauchy sequence of vectors always converges to a well-defined
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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.
Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "
Fréchet space."
Banach spaces originally grew out of the study of
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s by
Hilbert,
Fréchet, and
Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, the spaces under study are often Banach spaces.
Definition
A Banach space is a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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 ...
A normed space is a pair
[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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
where the distinction between the categories of normed spaces, normable spaces, metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
s, TVSs, topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s, etc. is usually important.
consisting of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a scalar field
(where
is commonly
or
) together with a distinguished
[This means that if the norm is replaced with a different norm then is the same normed space as even if the norms are equivalent. However, equivalence of norms on a given vector space does form an ]equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
.
norm Like all norms, this norm induces a
translation invariant
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equa ...
[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.]
distance function, called the canonical or
(norm) induced metric, defined by
[Because for all it is always true that for all So the order of and in this definition does not matter.]
for all vectors
This makes
into a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
A sequence
is called or or if for every real
there exists some index
such that
whenever
and
are greater than
The canonical metric
is called a if the pair
is a , which by definition means for every
in
there exists some
such that
where because
this sequence's convergence to
can equivalently be expressed as:
By definition, the normed space
is a if the norm induced metric
is a
complete metric
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in .
Intuitively, a space is complete if ther ...
, or said differently, if
is a
complete metric space.
The norm
of a normed space
is called a if
is a Banach space.
L-semi-inner product
For any normed space
there exists an
L-semi-inner product
In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or ...
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 product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s, which are what fundamentally distinguish
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 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
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.
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. The norm
is always a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
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
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
but a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
ball/
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equa ...
, which means that for any
and
the subset
is
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
(respectively,
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
) 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
is of the form
for some integer
(the closed ball can also be used instead of the open ball, although the indexing set
and radii
may need to be changed).
Additionally,
can always be chosen to be
countable if
is a , which by definition means that
contains some countable
dense subset.
The
Anderson–Kadec theorem states that every infinite–dimensional separable
Fréchet space is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to the
product space of countably many copies of
(this homeomorphism need not be a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
).
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including the separable
Hilbert 2 sequence space with its usual norm
where (in sharp contrast to finite−dimensional spaces)
is also
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to its
unit
There is a compact subset
of
whose
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
is closed and thus also compact (see this footnote
[Let be the separable ]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 ...
of square-summable sequences with the usual norm and let be the standard orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
(that is at the -coordinate). The closed set is compact (because it is sequentially compact) but its convex hull is 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
Hausdorff locally convex spaces, the convex hull of this compact subset is compact. The vector subspace is a pre-Hilbert space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often d ...
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 to be compact (although it will be precompact/totally bounded). for an example).
However, like in all Banach spaces, the
convex hull of this (and every other) compact subset will be compact. But if a normed space is not complete then it is in general guaranteed that
will be compact whenever
is; an example
can even be found in a (non-complete)
pre-Hilbert vector subspace of
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 a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is associated with 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 , which by definition refers to any TVS whose its topology is induced by some (possibly unknown)
norm.
Comparison of complete metrizable vector topologies
The
open mapping theorem implies that if
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
).
So for example, if
are Banach spaces with topologies
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
if they induce the same topology;
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 equivalent to that Banach space's given norm.
[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 L1-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 a Banach space since its completion is the proper superset Because holds on the map is continuous. Despite this, the norm is equivalent to the norm (because is complete but is not).]
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.
Complete norms vs complete metrics
A metric
on a vector space
is induced by a norm on
if and only if
is
translation invariant
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equa ...
and , 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
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
on
such that the topology that
induces on
is equal to
If
is
translation invariant
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equa ...
then
is a Banach space if and only if
is a complete metric space.
If
is translation invariant, then it may be possible for
to be a Banach space but for
to be a complete metric space (see this footnote
[The ]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 ...
is a Banach space where the absolute value is a norm on the real line that induces the usual Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
on Define a metric on by for all Just like induced metric, the metric also induces the usual Euclidean topology on However, is not a complete metric because the sequence defined by is a sequence but it does not converge to any point of As a consequence of not converging, this sequence cannot be a Cauchy sequence in (that is, it is not a Cauchy sequence with respect to the norm ) because if it was then the fact that is a Banach space would imply that it converges (a contradiction). for an example). In contrast, a theorem of Klee,
[The statement of the theorem is: Let be 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
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
. which also applies to all
metrizable topological vector spaces, implies that if there exists
[This metric is assumed to be translation-invariant. So in particular, this metric does even have to be induced by a norm.] complete metric
on
that induces the norm topology
on
then
is a Banach space.
A
Fréchet space is a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
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
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
).
However, the topology of every Fréchet space is induced by some
countable family of real-valued (necessarily continuous) maps called
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s, 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 (such norms would necessarily be continuous)
[A norm (or ]seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
) 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
but to not be a Banach/
normable space because its topology can not be defined by any 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
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
(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 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 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 complete TVS, meaning that it is enough to check that every Cauchy 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 (possibly unknown) norm (such spaces are called and
they are characterized by being Hausdorff and having a
bounded convex neighborhood of the origin), 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
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 ...
is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
if and only if its
strong dual space is normable, in which case
is a Banach space (
denotes the
strong dual space of
whose topology is a generalization of 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 ...
-induced topology on the
continuous dual space ; see this footnote
[ 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 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 ...
. In this way, the strong topology is a generalization of the usual dual norm-induced topology on for more details).
If
is a
metrizable locally convex TVS, then
is normable if and only if
is a
Fréchet–Urysohn space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X.
Fréchet–Urysohn spaces are a spec ...
.
[Gabriyelyan, S.S]
"On topological spaces and topological groups with certain local countable networks
(2014)
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 Banach space, where this Banach space is called a
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
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
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 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
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
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
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
between two Banach spaces to only the
short map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
These maps are the morphisms in the category of metric spaces, Met (Isbell 1 ...
s; in that case the space
reappears as a natural
bifunctor.
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
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
, 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
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
. 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
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
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
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
is continuous, which happens if and only if
is an open subset of
[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 real and ]
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 Real and imaginary parts of a linear functional, 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 In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
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, such as
which correspond (respectively) to the
coproduct and
product in the category of Banach spaces and short maps (discussed above).
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
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of a normed space
there is a natural norm on the quotient space
The quotient
is a Banach space when
is complete.
[see pp. 17–19 in .] 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 In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space X, is a vector subspace M for which there exists some other vector subspace N of X, called its (topological) complement in X, such th ...
of
if
is the
range of a
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 ...
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
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 of Banach spaces include: 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
and their special cases, the
sequence spaces that consist of scalar sequences indexed by
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s
; 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
For every separable Banach space
there is a closed subspace
of
such that
Any
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 ...
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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, linear in its first argument that satisfies the following:
For example, the space
is a Hilbert space.
The
Hardy spaces, the
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s are examples of Banach spaces that are related to
spaces and have additional structure. They are important in different branches of analysis,
Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
and
Partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s 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
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 ...
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 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), 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.
Since
is a Banach space (using the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
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.
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.
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 set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s in a real Banach space, one of them open, can be separated by a closed
affine hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
.
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.
A subset
in a Banach space
is total if the
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
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
[see p. 19 in .]
If
is a closed linear subspace in
one can associate the
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
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.
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
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
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.
A common p ...
.
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.
[see Theorem 2.6.23, p. 231 in .]
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.
The subset
of
consisting of non-negative measures of mass 1 (
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
s) is a convex w*-closed subset of the unit ball of
The
extreme point
In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex ...
s of
are the
Dirac measures on
The set of Dirac measures on
equipped with the w*-topology, is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to
The result has been extended by Amir and Cambern to the case when the multiplicative
Banach–Mazur distance between
and
is
The theorem is no longer true when the distance is
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
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.
The Hausdorff compact space
here is again the maximal ideal space, also called the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of
in the C*-algebra context.
Bidual
If
is a normed space, the (continuous) dual
of the dual
is called , or of
For every normed space
there is a natural map,