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In
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a sequence space is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
whose elements are infinite
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

sequence
s of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. Equivalently, it is a
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

natural numbers
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 grassl ...
''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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s with elements in ''K'', and can be turned into a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s of this space. Sequence spaces are typically equipped with a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, or at least the structure of a
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The most important sequence spaces in analysis are the ℓ''p'' spaces, consisting of the ''p''-power summable sequences, with the ''p''-norm. These are special cases of L''p'' spaces for the
counting measureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
on the set of natural numbers. Other important classes of sequences like
convergent sequence As the positive integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
s or null sequences form sequence spaces, respectively denoted ''c'' and ''c''0, with the
sup norm Image:Vector norm sup.svg, frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real number, real- or complex number, ...
. Any sequence space can also be equipped with the
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topology
of
pointwise convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, under which it becomes a special kind of
Fréchet space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
called FK-space.


Definition

A
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

sequence
x_ = \left(x_n\right)_ in a set X is just an X-valued map x_ : \N \to X whose value at n \in \N is denoted by x_n instead of the usual parentheses notation x(n).


Space of all sequences

Let \mathbb denote the field either of real or complex numbers. The product \mathbb^ denotes the set of all sequences of scalars in \mathbb. This set can become a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
when
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

vector addition
is defined by :\left(x_n\right)_ + \left(y_n\right)_ \stackrel \left(x_n + y_n\right)_ and the
scalar multiplication 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

scalar multiplication
is defined by :\alpha\left(x_n\right)_ \stackrel \left(\alpha x_n\right)_. A sequence space is any
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of \mathbb^. As a topological space, \mathbb^ is naturally endowed with the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a market Market may refer to: *Market (economics) *Market economy *Mark ...
. Under this topology, \mathbb^ is Fréchet, meaning that it is a complete,
metrizable In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
, locally convex
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(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 ga ...
norms on \mathbb^ (and thus the product topology cannot be defined by any
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
). Among Fréchet spaces, \mathbb^ is minimal in having no continuous norms: But the product topology is also unavoidable: \mathbb^ 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of interest, and endowing it with a topology ''different'' from the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

subspace topology
.


''p'' spaces

For 0 < p < \infty, \ell^p is the subspace of \mathbb^ consisting of all sequences x_ = \left(x_n\right)_ satisfying :\sum_n , x_n, ^p < \infty. If p \geq 1, then the real-valued operation \, \cdot\, _p defined by :\, x\, _p = \left(\sum_n, x_n, ^p\right)^ defines a norm on \ell^p. In fact, \ell^p is a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in or, alternatively, if every Cauchy sequence in converges in . Intuitively, a space is complet ...
with respect to this norm, and therefore is a
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. If 0 < p < 1, then \ell^p does not carry a norm, but rather a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
defined by :d(x,y) = \sum_n \left, x_n - y_n\^p.\, If p = \infty, then \ell^ is defined to be the space of all
bounded sequence Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded em ...
s endowed with the norm :\, x\, _\infty = \sup_n , x_n, , \ell^ is also a Banach space.


''c'', ''c''0 and ''c''00

The space of
convergent sequence As the positive integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
s ''c'' is a sequence space. This consists of all x_ \in \mathbb^ such that lim''n''→∞ ''x''''n'' exists. Since every convergent sequence is bounded, ''c'' is a linear subspace of 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 ''c''0 consists of all sequences whose limit is zero. This is a closed subspace of ''c'', and so again a Banach space. The subspace of eventually zero sequences ''c''00 consists 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 \left(x_\right)_ where x_ = 1/k for the first n entries (for and is zero everywhere else (i.e. is
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...
, but does not converge to a sequence in ''c''00.


Space of all finite sequences

Let :\mathbb^=\left\ , denote the space of finite sequences over \mathbb. As a vector space, \mathbb^ is equal to c_, but \mathbb^ has a different topology. For every
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
let \mathbb^n denote the usual
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
endowed with the
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
and let \operatorname_ : \mathbb^n \to \mathbb^ denote the canonical inclusion :\operatorname_\left(x_1, \ldots, x_n\right) = \left(x_1, \ldots, x_n, 0, 0, \ldots \right). The
image An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows ...
of each inclusion is :\operatorname \left( \operatorname_ \right) = \left\ = \mathbb^n \times \left\ and consequently, :\mathbb^ = \bigcup_ \operatorname \left( \operatorname_ \right). This family of inclusions gives \mathbb^ a
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, ...
\tau^, defined to be the finest topology on \mathbb^ such that all the inclusions are continuous (an example of a coherent topology). With this topology, \mathbb^ becomes a complete,
Hausdorff
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 spa ...
,
sequential In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
that is Fréchet–Urysohn. The topology \tau^ is also strictly finer than the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

subspace topology
induced on \mathbb^ by \mathbb^. Convergence in \tau^ has a natural description: if v \in \mathbb^ and v_ is a sequence in \mathbb^ then v_ \to v in \tau^ if and only v_ is eventually contained in a single image \operatorname \left( \operatorname_ \right) and v_ \to v under the natural topology of that image. Often, each image \operatorname \left( \operatorname_ \right) is identified with the corresponding \mathbb^n; explicitly, the elements \left( x_1, \ldots, x_n \right) \in \mathbb^n and \left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right) are identified. This is facilitated by the fact that the subspace topology on \operatorname \left( \operatorname_ \right), the
quotient topology as the quotient space of a disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-sha ...
from the map \operatorname_, and the Euclidean topology on \mathbb^n all coincide. With this identification, \left( \left(\mathbb^, \tau^\right), \left(\operatorname_\right)_\right) is the
direct limit In , 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 , , or in general objects from any . The way they are put together is specifi ...
of the directed system \left( \left(\mathbb^n\right)_, \left(\operatorname_\right)_,\N \right), where every inclusion adds trailing zeros: :\operatorname_\left(x_1, \ldots, x_m\right) = \left(x_1, \ldots, x_m, 0, \ldots, 0 \right). This shows \left(\mathbb^, \tau^\right) is an LB-space.


Other sequence spaces

The space of bounded
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
, denote by bs, is the space of sequences x for which :\sup_n \left\vert \sum_^n x_i \right\vert < \infty. This space, when equipped with the norm :\, x\, _ = \sup_n \left\vert \sum_^n x_i \right\vert, is a Banach space isometrically isomorphic to \ell^, via the
linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
:(x_n)_ \mapsto \left(\sum_^n x_i\right)_. 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 c_ 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 The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
in many sequence spaces.


Properties of ℓ''p'' spaces and the space ''c''0

The space ℓ2 is the only ℓ''p'' space that is a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, since any norm that is induced by an
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
should satisfy the
parallelogram law A parallelogram. The sides are shown in blue and the diagonals in red. 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 ...

parallelogram law
:\, x+y\, _p^2 + \, x-y\, _p^2= 2\, x\, _p^2 + 2\, y\, _p^2. Substituting two distinct unit vectors for ''x'' and ''y'' directly shows that the identity is not true unless ''p'' = 2. Each ℓ''p'' is distinct, in that ℓ''p'' is a strict
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
of ℓ''s'' whenever ''p'' < ''s''; furthermore, ℓ''p'' is not linearly
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to ℓ''s'' when ''p'' ≠ ''s''. In fact, by Pitt's theorem , every bounded linear operator from ℓ''s'' to ℓ''p'' is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
when ''p'' < ''s''. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓ''s'', and is thus said to be
strictly singularIn functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional an ...
. If 1 < ''p'' < ∞, then the (continuous) dual space of ℓ''p'' is isometrically isomorphic to ℓ''q'', where ''q'' is the Hölder conjugate of ''p'': 1/''p'' + 1/''q'' = 1. The specific isomorphism associates to an element ''x'' of ℓ''q'' the functional :L_x(y) = \sum_n x_ny_n for ''y'' in ℓ''p''.
Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality Inequality may refer to: Economics * Attention inequality Attention inequality is a term used to target the inequality of distribution of atte ...
implies that ''L''''x'' is a bounded linear functional on ℓ''p'', and in fact :, L_x(y), \le \, x\, _q\,\, y\, _p so that the operator norm satisfies :\, L_x\, _ \stackrel\sup_ \frac \le \, x\, _q. In fact, taking ''y'' to be the element of ℓ''p'' with :y_n = \begin0&\rm\ x_n=0\\ x_n^, x_n, ^q &\rm\ x_n\not=0 \end gives ''L''''x''(''y'') = , , ''x'', , ''q'', so that in fact :\, L_x\, _ = \, x\, _q. Conversely, given a bounded linear functional ''L'' on ℓ''p'', the sequence defined by ''x''''n'' = ''L''(''e''''n'') lies in ℓ''q''. Thus the mapping x\mapsto L_x gives an isometry :\kappa_q : \ell^q \to (\ell^p)^*. The map :\ell^q\xrightarrow(\ell^p)^*\xrightarrow obtained by composing κ''p'' with the inverse of its
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...
coincides with the
canonical injection
canonical injection
of ℓ''q'' into its double dual. As a consequence ℓ''q'' is a
reflexive space Reflexive may refer to: In fiction: *Metafiction In grammar: *Reflexive pronoun, a pronoun with a reflexive relationship with its self-identical antecedent *Reflexive verb, where a semantic agent and patient are the same In mathematics and comput ...
. By
abuse of notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of ''c''0 is ℓ1; the dual of ℓ1 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. The spaces ''c''0 and ℓ''p'' (for 1 ≤ ''p'' < ∞) have a canonical unconditional
Schauder basisIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, where ''e''''i'' is the sequence which is zero but for a 1 in the ''i'' th 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 In mathematics, weak topology is an alternative term for certain initial topology, 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 initia ...
on infinite-dimensional spaces is strictly weaker than the
strong topologyIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, there are nets in ℓ1 that are weak convergent but not strong convergent. The ℓ''p'' spaces can be embedded into many
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 spaceIn mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an lp space, ℓ ''p'' space nor a Sequence space#c and c0, ''c''0 space can be embedded. The Tsirelson space is refle ...
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 Q:\ell^1 \to X, so that ''X'' is isomorphic to \ell^1 / \ker Q. In general, ker ''Q'' is not complemented in ℓ1, that is, there does not exist a subspace ''Y'' of ℓ1 such that \ell^1 = Y \oplus \ker Q. In fact, ℓ1 has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take X=\ell^p; 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.


''p'' spaces are increasing in ''p''

For p\in ,\infty/math>, the spaces \ell^p are increasing in p, with the inclusion operator being continuous: for 1\le p, one has \, f\, _q\le\, f\, _p. This follows from defining F:=\frac for f\in\ell^p, and noting that , F(m), \le1 for all m\in\mathbb, which can be shown to imply \, F\, _q^q\le1.


''2'' is isomorphic to all separable, infinite dimensional Hilbert spaces

Let H be a separable Hilbert space. Every orthogonal set in H countable (i.e. has finite
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
or \,\aleph_0\,). The following two items are related: * If H is infinite dimensional, then it is isomorphic to ℓ''2'' * If ''dim(H) = N'', then H is isomorphic to \Complex''N''


Properties of ℓ1 spaces

A sequence of elements in ℓ1 converges in the space of complex sequences ℓ1 if and only if it converges weakly in this space. If ''K'' is a subset of this space, then the following are equivalent: # ''K'' is compact; # ''K'' is weakly compact; # ''K'' is bounded, closed, and equismall at infinity. Here ''K'' being equismall at infinity means that for every \epsilon > 0, there exists a natural number n_ \geq 0 such that \sum_^ , s_n , < \epsilon for all s = \left( s_n \right)_^ \in K.


See also

* Lp space *
Tsirelson spaceIn mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an lp space, ℓ ''p'' space nor a Sequence space#c and c0, ''c''0 space can be embedded. The Tsirelson space is refle ...
* beta-dual space * Orlicz sequence space *
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...


References


Bibliography

* . * . * * . * * * . * {{Authority control Functional analysis Sequences and series