In ^{''p''} spaces, consisting of the ''p''-power summable sequences, with the ''p''-norm. These are special cases of L^{''p''} spaces for the _{0}, with the

_{''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 $\backslash left(x\_\backslash right)\_$ where $x\_\; =\; 1/k$ for the first $n$ entries (for and is zero everywhere else (i.e. is _{00}.

^{2} is the only ℓ^{''p''} space that is a ^{''p''} is distinct, in that ℓ^{''p''} is a strict ^{''s''} whenever ''p'' < ''s''; furthermore, ℓ^{''p''} is not linearly ^{''s''} when ''p'' ≠ ''s''. In fact, by Pitt's theorem , every bounded linear operator from ℓ^{''s''} to ℓ^{''p''} is ^{''s''}, and is thus said to be ^{''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
:$L\_x(y)\; =\; \backslash sum\_n\; x\_ny\_n$
for ''y'' in ℓ^{''p''}. _{''x''} is a bounded linear functional on ℓ^{''p''}, and in fact
:$,\; L\_x(y),\; \backslash le\; \backslash ,\; x\backslash ,\; \_q\backslash ,\backslash ,\; y\backslash ,\; \_p$
so that the operator norm satisfies
:$\backslash ,\; L\_x\backslash ,\; \_\; \backslash stackrel\backslash sup\_\; \backslash frac\; \backslash le\; \backslash ,\; x\backslash ,\; \_q.$
In fact, taking ''y'' to be the element of ℓ^{''p''} with
:$y\_n\; =\; \backslash begin0\&\backslash rm\backslash \; x\_n=0\backslash \backslash \; x\_n^,\; x\_n,\; ^q\; \&\backslash rm\backslash \; x\_n\backslash not=0\; \backslash end$
gives ''L''_{''x''}(''y'') = , , ''x'', , _{''q''}, so that in fact
:$\backslash ,\; L\_x\backslash ,\; \_\; =\; \backslash ,\; x\backslash ,\; \_q.$
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 $x\backslash mapsto\; L\_x$ gives an isometry
:$\backslash kappa\_q\; :\; \backslash ell^q\; \backslash to\; (\backslash ell^p)^*.$
The map
:$\backslash ell^q\backslash xrightarrow(\backslash ell^p)^*\backslash xrightarrow$
obtained by composing κ_{''p''} with the inverse of its ^{''q''} into its double dual. As a consequence ℓ^{''q''} is a ^{''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 _{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.
The spaces ''c''_{0} and ℓ^{''p''} (for 1 ≤ ''p'' < ∞) have a canonical unconditional _{''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 ^{1} that are weak convergent but not strong convergent.
The ℓ^{''p''} spaces can be embedded into many ^{''p''} or of ''c''_{0}, was answered negatively by B. S. Tsirelson's construction of ^{1}, was answered in the affirmative by . That is, for every separable Banach space ''X'', there exists a quotient map $Q:\backslash ell^1\; \backslash to\; X$, so that ''X'' is isomorphic to $\backslash ell^1\; /\; \backslash ker\; Q$. In general, ker ''Q'' is not complemented in ℓ^{1}, that is, there does not exist a subspace ''Y'' of ℓ^{1} such that $\backslash ell^1\; =\; Y\; \backslash oplus\; \backslash ker\; Q$. In fact, ℓ^{1} has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take $X=\backslash ell^p$; 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.

^{''2''}
* If ''dim(H) = N'', then H is isomorphic to $\backslash Complex$^{''N''}

^{1} converges in the space of complex sequences ℓ^{1} 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 $\backslash epsilon\; >\; 0$, there exists a natural number $n\_\; \backslash geq\; 0$ such that $\backslash sum\_^\; ,\; s\_n\; ,\; <\; \backslash epsilon$ for all $s\; =\; \backslash left(\; s\_n\; \backslash right)\_^\; \backslash in\; K$.

^{p} space
*

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 ...

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 ...

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 ℓcounting 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''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 ...

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

Asequence
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 ...

$x\_\; =\; \backslash left(x\_n\backslash right)\_$ in a set $X$ is just an $X$-valued map $x\_\; :\; \backslash N\; \backslash to\; X$ whose value at $n\; \backslash in\; \backslash N$ is denoted by $x\_n$ instead of the usual parentheses notation $x(n).$
Space of all sequences

Let $\backslash mathbb$ denote the field either of real or complex numbers. The product $\backslash mathbb^$ denotes the set of all sequences of scalars in $\backslash mathbb.$ This set can become avector 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 ...

is defined by
:$\backslash left(x\_n\backslash right)\_\; +\; \backslash left(y\_n\backslash right)\_\; \backslash stackrel\; \backslash left(x\_n\; +\; y\_n\backslash right)\_$
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 ...

is defined by
:$\backslash alpha\backslash left(x\_n\backslash right)\_\; \backslash stackrel\; \backslash left(\backslash alpha\; x\_n\backslash right)\_.$
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 $\backslash mathbb^.$
As a topological space, $\backslash mathbb^$ 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, $\backslash mathbb^$ 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 $\backslash mathbb^$ (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, $\backslash mathbb^$ is minimal in having no continuous norms:
But the product topology is also unavoidable: $\backslash mathbb^$ 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 ...

.
ℓ^{''p''} spaces

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 $\backslash ell^p$ 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
:$d(x,y)\; =\; \backslash sum\_n\; \backslash left,\; x\_n\; -\; y\_n\backslash ^p.\backslash ,$
If $p\; =\; \backslash infty,$ then $\backslash ell^$ 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
:$\backslash ,\; x\backslash ,\; \_\backslash infty\; =\; \backslash sup\_n\; ,\; x\_n,\; ,$
$\backslash ell^$ is also a Banach space.
''c'', ''c''_{0} and ''c''_{00}

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 $x\_\; \backslash in\; \backslash mathbb^$ such that limCauchy
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''Space of all finite sequences

Let :$\backslash mathbb^=\backslash left\backslash $, denote the space of finite sequences over $\backslash mathbb$. As a vector space, $\backslash mathbb^$ is equal to $c\_$, but $\backslash mathbb^$ has a different topology. For everynatural 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 $\backslash mathbb^n$ 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 $\backslash operatorname\_\; :\; \backslash mathbb^n\; \backslash to\; \backslash mathbb^$ denote the canonical inclusion
:$\backslash operatorname\_\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; \backslash ldots\; \backslash right)$.
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
:$\backslash operatorname\; \backslash left(\; \backslash operatorname\_\; \backslash right)\; =\; \backslash left\backslash \; =\; \backslash mathbb^n\; \backslash times\; \backslash left\backslash $
and consequently,
:$\backslash mathbb^\; =\; \backslash bigcup\_\; \backslash operatorname\; \backslash left(\; \backslash operatorname\_\; \backslash right).$
This family of inclusions gives $\backslash mathbb^$ 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, ...

$\backslash tau^$, defined to be the finest topology on $\backslash mathbb^$ such that all the inclusions are continuous (an example of a coherent topology). With this topology, $\backslash mathbb^$ becomes a complete, , 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 $\backslash tau^$ 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 ...

induced on $\backslash mathbb^$ by $\backslash mathbb^$.
Convergence in $\backslash tau^$ has a natural description: if $v\; \backslash in\; \backslash mathbb^$ and $v\_$ is a sequence in $\backslash mathbb^$ then $v\_\; \backslash to\; v$ in $\backslash tau^$ if and only $v\_$ is eventually contained in a single image $\backslash operatorname\; \backslash left(\; \backslash operatorname\_\; \backslash right)$ and $v\_\; \backslash to\; v$ under the natural topology of that image.
Often, each image $\backslash operatorname\; \backslash left(\; \backslash operatorname\_\; \backslash right)$ is identified with the corresponding $\backslash mathbb^n$; explicitly, the elements $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n\; \backslash right)\; \backslash in\; \backslash mathbb^n$ and $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; 0,\; \backslash ldots\; \backslash right)$ are identified. This is facilitated by the fact that the subspace topology on $\backslash operatorname\; \backslash left(\; \backslash operatorname\_\; \backslash right)$, 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 $\backslash operatorname\_$, and the Euclidean topology on $\backslash mathbb^n$ all coincide. With this identification, $\backslash left(\; \backslash left(\backslash mathbb^,\; \backslash tau^\backslash right),\; \backslash left(\backslash operatorname\_\backslash right)\_\backslash right)$ 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 $\backslash left(\; \backslash left(\backslash mathbb^n\backslash right)\_,\; \backslash left(\backslash operatorname\_\backslash right)\_,\backslash N\; \backslash right),$ where every inclusion adds trailing zeros:
:$\backslash operatorname\_\backslash left(x\_1,\; \backslash ldots,\; x\_m\backslash right)\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_m,\; 0,\; \backslash ldots,\; 0\; \backslash right)$.
This shows $\backslash left(\backslash mathbb^,\; \backslash tau^\backslash right)$ is an LB-space.
Other sequence spaces

The space of boundedseries
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
:$\backslash sup\_n\; \backslash left\backslash vert\; \backslash sum\_^n\; x\_i\; \backslash right\backslash vert\; <\; \backslash infty.$
This space, when equipped with the norm
:$\backslash ,\; x\backslash ,\; \_\; =\; \backslash sup\_n\; \backslash left\backslash vert\; \backslash sum\_^n\; x\_i\; \backslash right\backslash vert,$
is a Banach space isometrically isomorphic to $\backslash ell^,$ 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 ...

:$(x\_n)\_\; \backslash mapsto\; \backslash left(\backslash sum\_^n\; x\_i\backslash right)\_.$
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}

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 ...

:$\backslash ,\; x+y\backslash ,\; \_p^2\; +\; \backslash ,\; x-y\backslash ,\; \_p^2=\; 2\backslash ,\; x\backslash ,\; \_p^2\; +\; 2\backslash ,\; y\backslash ,\; \_p^2.$
Substituting two distinct unit vectors for ''x'' and ''y'' directly shows that the identity is not true unless ''p'' = 2.
Each ℓsubset
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). ...

of ℓisomorphic
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 ...

to ℓcompact
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 ℓstrictly 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 ℓHölder'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''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 of ℓreflexive 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 ℓdual
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''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''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 ℓBanach 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 ℓTsirelson 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 ℓ ℓ^{''p''} spaces are increasing in ''p''

ℓ^{''2''} is isomorphic to all separable, infinite dimensional Hilbert spaces

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 $\backslash ,\backslash aleph\_0\backslash ,$). The following two items are related:
* If H is infinite dimensional, then it is isomorphic to ℓ Properties of ℓ^{1} spaces

See also

* LTsirelson 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