Sequence space
Encyclopedia
In functional analysis
and related areas of mathematics
, a sequence space is a vector space
whose elements are infinite sequence
s of real
or complex numbers. Equivalently, it is a function space
whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space
under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspace
s of this space. Sequence spaces are typically equipped with a norm
, or at least the structure of a topological vector space
.
The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces
for the counting measure
on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology
of pointwise convergence
, under which it becomes a special kind of Fréchet space
called FK-space
.
This can be turned into a vector space
by defining vector addition as
and the scalar multiplication
as
A sequence space is any linear subspace of KN.
If p ≥ 1, then the real-valued operation
defined by
defines a norm
on ℓp. In fact, ℓp is a complete metric space with respect to this norm, and therefore is a Banach space
.
If 0 < p < 1, then ℓp does not carry a norm, but rather a metric
defined by
If p = ∞, then ℓ∞ is defined to be the space of all bounded sequences. With respect to the norm
ℓ∞ is also a Banach space.
s c is a sequence space. This consists of all x ∈ KN such that limn→∞xn exists. Since every convergent sequence is bounded, c is a linear subspace of ℓ∞. It is, moreover, a closed subspace with respect to the infinity norm, and so a Banach space in its own right.
The subspace of null sequences c0 consists of all sequences whose limit is zero. This is a closed subspace of c, and so again a Banach space.
, denote by bs, is the space of sequences x for which
This space, when equipped with the norm
is a Banach space isometrically isomorphic to ℓ∞, via the linear mapping
The subspace cs consisting of all convergent series is a subspace that goes over to the space c under this isomorphism.
The space Φ 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
in many sequence spaces.
, since any norm that is induced by an inner product should satisfy the parallelogram identity
. Substituting two distinct unit vectors in for x and y directly shows that the identity is not true unless p = 2.
Each ℓp is distinct, in that ℓp is a strict subset of ℓs whenever p < s; furthermore, ℓp is not linearly isomorphic to ℓs when p ≠ s. In fact, by Pitt's theorem , every bounded linear operator from ℓs to ℓp is compact
when p < s. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓs, and is thus said to be strictly singular.
If 1 < p < ∞, then the (continuous) dual space
of ℓp is isometrically isomorphic to ℓq, where q is the Hölder conjugate of p: 1/p + 1/q = 1. The specific isomorphism associates to an element x of ℓq the functional
for y in ℓp. Hölder's inequality
implies that Lx is a bounded linear functional on ℓp, and in fact
so that the operator norm satisfies
In fact, taking y to be the element of ℓp with
gives Lx(y) = ||x||q, so that in fact
Conversely, given a bounded linear functional L on ℓp, the sequence defined by xn = L(en) lies in ℓq. Thus the mapping
gives an isometry
The map
obtained by composing κp with the inverse of its transpose coincides with the canonical injection of ℓq into its double dual. As a consequence ℓq is a reflexive space
. By abuse of notation
, it is typical to identify ℓq with the dual of ℓp: (ℓp)* = ℓq. Then reflexivity is understood by the sequence of identifications (ℓp)** = (ℓq)* = ℓp.
The space c0 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 c0 is ℓ1; the dual of ℓ1 is ℓ∞. For the case of natural numbers index set, the ℓp and c0 are separable, with the sole exception of ℓ∞. The dual of ℓ∞ is the ba space
.
The spaces c0 and ℓp (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis
{ei | i = 1, 2,…}, where ei is the sequence which is zero but for a 1 in the ith entry.
The space ℓ1 has the Schur property
: In ℓ1, any sequence that is weakly convergent
is also strongly convergent
. However, since the weak topology
on infinite-dimensional spaces is strictly weaker than the strong topology
, there are nets
in ℓ1 that are weak convergent but not strong convergent.
The ℓp spaces can be embedded
into many Banach space
s. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓp or of c0, 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 ℓ1, 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 ℓ1, that is, there does not exist a subspace Y of ℓ1 such that 
. In fact, ℓ1 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 ℓp is isomorphic to any other, there are thus uncountably many ker Q 's).
Except for the trivial finite dimensional case, an unusual feature of ℓp is that it is not polynomially reflexive
.
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a sequence space is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
whose elements are infinite sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s of real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex numbers. Equivalently, it is a function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
s of this space. Sequence spaces are typically equipped with a norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
, or at least the structure of a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
.
The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
for the counting measure
Counting measure
In mathematics, 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 is finite, and ∞ if the subset is infinite....
on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of pointwise convergence
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
, under which it becomes a special kind of Fréchet space
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...
called FK-space
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-spaces....
.
Definition
Let K denote the field either of real or complex numbers. Denote by KN the set of all sequences of scalarsThis can be turned into a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
by defining vector addition as
and the scalar multiplication
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
as
A sequence space is any linear subspace of KN.
ℓp spaces
For 0 < p < ∞, ℓp is the subspace of KN consisting of all sequences x = (xn) satisfyingIf p ≥ 1, then the real-valued operation
defined bydefines a norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on ℓp. In fact, ℓp is a complete metric space with respect to this norm, and therefore is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
.
If 0 < p < 1, then ℓp does not carry a norm, but rather a metric
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
defined by
If p = ∞, then ℓ∞ is defined to be the space of all bounded sequences. With respect to the norm
ℓ∞ is also a Banach space.
c and c0
The space of convergent sequenceLimit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
s c is a sequence space. This consists of all x ∈ KN such that limn→∞xn exists. Since every convergent sequence is bounded, c is a linear subspace of ℓ∞. It is, moreover, a closed subspace with respect to the infinity norm, and so a Banach space in its own right.
The subspace of null sequences c0 consists of all sequences whose limit is zero. This is a closed subspace of c, and so again a Banach space.
Other sequence spaces
The space of bounded seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
, denote by bs, is the space of sequences x for which
This space, when equipped with the norm
is a Banach space isometrically isomorphic to ℓ∞, via the linear mapping
The subspace cs consisting of all convergent series is a subspace that goes over to the space c under this isomorphism.
The space Φ 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
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in many sequence spaces.
Properties of ℓp spaces and the space c0
The space ℓ2 is the only ℓp space that is a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, since any norm that is induced by an inner product should satisfy the parallelogram identity
. Substituting two distinct unit vectors in for x and y directly shows that the identity is not true unless p = 2.Each ℓp is distinct, in that ℓp is a strict subset of ℓs whenever p < s; furthermore, ℓp is not linearly isomorphic to ℓs when p ≠ s. In fact, by Pitt's theorem , every bounded linear operator from ℓs to ℓp is compact
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
when p < s. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓs, and is thus said to be strictly singular.
If 1 < p < ∞, then the (continuous) dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of ℓp is isometrically isomorphic to ℓq, where q is the Hölder conjugate of p: 1/p + 1/q = 1. The specific isomorphism associates to an element x of ℓq the functional
for y in ℓp. Hölder's inequality
Hölder's inequality
In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
implies that Lx is a bounded linear functional on ℓp, and in fact
so that the operator norm satisfies
In fact, taking y to be the element of ℓp with
gives Lx(y) = ||x||q, so that in fact
Conversely, given a bounded linear functional L on ℓp, the sequence defined by xn = L(en) lies in ℓq. Thus the mapping
gives an isometryThe map
obtained by composing κp with the inverse of its transpose coincides with the canonical injection of ℓq into its double dual. As a consequence ℓq is a reflexive space
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
. By abuse of notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
, it is typical to identify ℓq with the dual of ℓp: (ℓp)* = ℓq. Then reflexivity is understood by the sequence of identifications (ℓp)** = (ℓq)* = ℓp.
The space c0 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
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of c0 is ℓ1; the dual of ℓ1 is ℓ∞. For the case of natural numbers index set, the ℓp and c0 are separable, with the sole exception of ℓ∞. The dual of ℓ∞ is the ba space
Ba space
In mathematics, the ba space ba of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive measures on \Sigma. The norm is defined as the variation, that is \|\nu\|=|\nu|....
.
The spaces c0 and ℓp (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis
Schauder basis
In mathematics, a Schauder basis or countable basis is similar to the usual 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...
{ei | i = 1, 2,…}, where ei is the sequence which is zero but for a 1 in the ith entry.
The space ℓ1 has the Schur property
Schur's property
In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm.-Motivation:...
: In ℓ1, any sequence that is weakly convergent
Weak convergence
In mathematics, weak convergence may refer to:* The weak convergence of random variables of a probability distribution.* The weak convergence of a sequence of probability measures....
is also strongly convergent
Strong convergence
In mathematics, strong convergence may refer to:* The strong convergence of random variables of a probability distribution.* The strong convergence of a sequence in a Hilbert space....
. However, since the weak topology
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
on infinite-dimensional spaces is strictly weaker than the strong topology
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...
, there are nets
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
in ℓ1 that are weak convergent but not strong convergent.
The ℓp spaces can be embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
into many Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓp or of c0, was answered negatively by B. S. Tsirelson
Boris Tsirelson
Boris Semyonovich Tsirelson is a Soviet-Israeli mathematician and Professor of Mathematics in the Tel Aviv University in Israel.-Biography:Boris Tsirelson was born in Leningrad to a Russian Jewish family...
's construction of Tsirelson space
Tsirelson space
In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l p space nor a c0 space can be embedded.It was introduced by B. S. Tsirelson in 1974...
in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
of ℓ1, 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 ℓ1, that is, there does not exist a subspace Y of ℓ1 such that . In fact, ℓ1 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 ℓp is isomorphic to any other, there are thus uncountably many ker Q 's).Except for the trivial finite dimensional case, an unusual feature of ℓp is that it is not polynomially reflexive
Polynomially reflexive space
In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space....
.