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 defined on ...
and related areas of
mathematics, a sequence space is 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 ca ...
whose elements are infinite
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex numbers
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 form a ...
. Equivalently, it is a
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 vec ...
whose elements are functions from the
natural numbers
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 ...
to the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' of real or complex numbers. The set of all such functions is naturally identified with the set of all possible
infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s with elements in ''K'', and can be turned into 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 ca ...
under the operations of
pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are
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, ...
s of this space. Sequence spaces are typically equipped with a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
, or at least the structure of a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
.
The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the ''p''-norm. These are special cases of
L''p'' spaces for the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infini ...
on the set of natural numbers. Other important classes of sequences like
convergent sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limi ...
s or
null sequences form sequence spaces, respectively denoted ''c'' and ''c''
0, with the
sup norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
. Any sequence space can also be equipped with the
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 h ...
of
pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
Suppose that X is a set and ...
, under which it becomes a special kind of
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to th ...
called
FK-space
In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with a normable topology are called BK-spa ...
.
Definition
A
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
in a set
is just an
-valued map
whose value at
is denoted by
instead of the usual parentheses notation
Space of all sequences
Let
denote the field either of real or complex numbers. The set
of all
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of elements of
is 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 ca ...
for
componentwise addition
:
and componentwise
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
:
A sequence space is any
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, ...
of
As a topological space,
is naturally endowed 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-seemi ...
. Under this topology,
is
Fréchet, meaning that it is a
complete,
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
,
locally convex topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
(TVS). However, this topology is rather pathological: there are no
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 ...
norms on
(and thus the product topology cannot
be defined by any
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
). Among Fréchet spaces,
is minimal in having no continuous norms:
But the product topology is also unavoidable:
does not admit a
strictly coarser Hausdorff, locally convex topology. For that reason, the study of sequences begins by finding a strict
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, ...
of interest, and endowing it with a topology ''different'' from the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
.
spaces
For
is the subspace of
consisting of all sequences
satisfying
If
then the real-valued function
on
defined by
defines a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
on
In fact,
is a
complete metric space with respect to this norm, and therefore is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
.
If
then
is also a
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 ...
when endowed with its canonical
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 ...
, called the , defined for all
by
The canonical norm induced by this inner product is the usual
-norm, meaning that
for all
If
then
is defined to be the space of all
bounded sequence
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. A ...
s endowed with the norm
is also a Banach space.
If
then
does not carry a norm, but rather 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 mathema ...
defined by
''c'', ''c''0 and ''c''00
A is any sequence
such that
exists.
The set of all convergent sequences is a vector subspace of
called the
. Since every convergent sequence is bounded,
is a linear subspace of
Moreover, this sequence space is a closed subspace of
with respect to the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
, and so it is a Banach space with respect to this norm.
A sequence that converges to
is called a and is said to . The set of all sequences that converge to
is a closed vector subspace of
that when endowed with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
becomes a Banach space that is denoted by and is called the or the .
The , is the subspace of
consisting of all sequences which have only finitely many nonzero elements. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm. For example, the sequence
where
for the first
entries (for
) and is zero everywhere else (that is,
) is a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
but it does not converge to a sequence in
Space of all finite sequences
Let
:
,
denote the space of finite sequences over
. As a vector space,
is equal to
, but
has a different topology.
For every
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 ...
let
denote the usual
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
endowed with the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
and let
denote the canonical inclusion
:
.
The
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of each inclusion is
:
and consequently,
:
This family of inclusions gives
a
final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that m ...
, defined to be the
finest topology on
such that all the inclusions are continuous (an example of a
coherent topology). With this topology,
becomes a
complete,
Hausdorff,
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
,
sequential
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
,
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
that is
Fréchet–Urysohn. The topology
is also
strictly finer than the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
induced on
by
.
Convergence in
has a natural description: if
and
is a sequence in
then
in
if and only
is eventually contained in a single image
and
under the natural topology of that image.
Often, each image
is identified with the corresponding
; explicitly, the elements
and
are identified. This is facilitated by the fact that the subspace topology on
, the
quotient topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
from the map
, and the Euclidean topology on
all coincide. With this identification,
is the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of the directed system
where every inclusion adds trailing zeros:
:
.
This shows
is an
LB-space In mathematics, an ''LB''-space, also written (''LB'')-space, is a topological vector space X that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Banach spaces.
This means that X is a direct limit of a direct syste ...
.
Other sequence spaces
The space of bounded
series, denote by
bs, is the space of sequences
for which
:
This space, when equipped with the norm
:
is a Banach space isometrically isomorphic to
via the
linear mapping
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 pre ...
:
The subspace ''cs'' consisting of all convergent series is a subspace that goes over to the space ''c'' under this isomorphism.
The space Φ or
is defined to be the space of all infinite sequences with only a finite number of non-zero terms (sequences with
finite support). This set is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
in many sequence spaces.
Properties of ℓ''p'' spaces and the space ''c''0
The space ℓ
2 is the only ℓ
''p'' space that is a
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 ...
, since any norm that is induced by an
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 ...
should satisfy the
parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the ...
:
Substituting two distinct unit vectors for ''x'' and ''y'' directly shows that the identity is not true unless ''p'' = 2.
Each is distinct, in that is a strict
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of whenever ''p'' < ''s''; furthermore, is not linearly
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to when . In fact, by Pitt's theorem , every bounded linear operator from to is
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 Briti ...
when . No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of , and is thus said to be
strictly singular In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator between normed spaces which is not bounded below on any infinite-dimensional subspace.
Definitions.
Let ''X'' and ''Y'' be normed linea ...
.
If 1 < ''p'' < ∞, then the
(continuous) dual space of ℓ
''p'' is isometrically isomorphic to ℓ
''q'', where ''q'' is the
Hölder conjugate In mathematics, two real numbers p, q>1 are called conjugate indices (or Hölder conjugates) if
: \frac + \frac = 1.
Formally, we also define q = \infty as conjugate to p=1 and vice versa
References
Additional references
*
*
{{Lat ...
of ''p'': 1/''p'' + 1/''q'' = 1. The specific isomorphism associates to an element ''x'' of the functional
for ''y'' in .
Hölder's inequality implies that ''L''
''x'' is a bounded linear functional on , and in fact
so that the operator norm satisfies
:
In fact, taking ''y'' to be the element of with
:
gives ''L''
''x''(''y'') = , , ''x'', ,
''q'', so that in fact
:
Conversely, given a bounded linear functional ''L'' on , the sequence defined by lies in ℓ
''q''. Thus the mapping
gives an isometry
The map
:
obtained by composing κ
''p'' with the inverse of its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
coincides with the
canonical injection of ℓ
''q'' into its
double dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
. As a consequence ℓ
''q'' is a
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an is ...
. By
abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
, it is typical to identify ℓ
''q'' with the dual of ℓ
''p'': (ℓ
''p'')
* = ℓ
''q''. Then reflexivity is understood by the sequence of identifications (ℓ
''p'')
** = (ℓ
''q'')
* = ℓ
''p''.
The space ''c''
0 is defined as the space of all sequences converging to zero, with norm identical to , , ''x'', ,
∞. It is a closed subspace of ℓ
∞, hence a Banach space. The
dual of ''c''
0 is ℓ
1; the dual of ℓ
1 is ℓ
∞. For the case of natural numbers index set, the ℓ
''p'' and ''c''
0 are
separable, with the sole exception of ℓ
∞. The dual of ℓ
∞ is the
ba space
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X).
If Σ i ...
.
The spaces ''c''
0 and ℓ
''p'' (for 1 ≤ ''p'' < ∞) have a canonical unconditional
Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
, where ''e''
''i'' is the sequence which is zero but for a 1 in the ''i''
th entry.
The space ℓ
1 has the
Schur property: In ℓ
1, any sequence that is
weakly convergent is also
strongly convergent . However, since the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on infinite-dimensional spaces is strictly weaker than the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:
* the final topology on the disjoint union
* the to ...
, there are
nets in ℓ
1 that are weak convergent but not strong convergent.
The ℓ
''p'' spaces can be
embedded into many
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ
''p'' or of ''c''
0, was answered negatively by
B. S. Tsirelson's construction of
Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a
quotient space of ℓ
1, was answered in the affirmative by . That is, for every separable Banach space ''X'', there exists a quotient map
, so that ''X'' is isomorphic to
. In general, ker ''Q'' is not complemented in ℓ
1, that is, there does not exist a subspace ''Y'' of ℓ
1 such that
. In fact, ℓ
1 has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take
; since there are uncountably many such ''X''s, and since no ℓ
''p'' is isomorphic to any other, there are thus uncountably many ker ''Q''s).
Except for the trivial finite-dimensional case, an unusual feature of ℓ
''p'' is that it is not
polynomially reflexive.
ℓ''p'' spaces are increasing in ''p''
For