TheInfoList

OR:

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 ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sequence space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
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 call ...
s of real 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 ...
. 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 ve ...
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 n ...
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 In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
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, or at least the structure of a topological vector space. 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 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 ...
of pointwise convergence, 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 t ...
called FK-space.

# 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 call ...
$x_ = \left\left(x_n\right\right)_$ 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 $x\left(n\right).$

## Space of all sequences

Let $\mathbb$ denote the field either of real or complex numbers. The set $\mathbb^$ 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 $\mathbb$ is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
for componentwise addition :$\left\left(x_n\right\right)_ + \left\left(y_n\right\right)_ = \left\left(x_n + y_n\right\right)_,$ and componentwise scalar multiplication :$\alpha\left\left(x_n\right\right)_ = \left\left(\alpha x_n\right\right)_.$ 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 $\mathbb^.$ As a topological space, $\mathbb^$ 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, $\mathbb^$ 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 (TVS). However, this topology is rather pathological: there are no continuous norms on $\mathbb^$ (and thus the product topology cannot be defined by any norm). 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, 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 $0 < p < \infty,$ $\ell^p$ is the subspace of $\mathbb^$ consisting of all sequences $x_ = \left\left(x_n\right\right)_$ satisfying $\sum_n , x_n, ^p < \infty.$ If $p \geq 1,$ then the real-valued function $\, \cdot\, _p$ on $\ell^p$ defined by $\, x\, _p ~=~ \left(\sum_n, x_n, ^p\right)^ \qquad \text x \in \ell^p$ 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 . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
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 vec ...
. If $p = 2$ then $\ell^2$ 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 de ...
, called the , defined for all $x_\bull, y_\bull \in \ell^p$ by $\langle x_\bull, y_\bull \rangle ~=~ \sum_n \overline y_n.$ The canonical norm induced by this inner product is the usual $\ell^2$-norm, meaning that $\, \mathbf\, _2 = \sqrt$ for all $\mathbf \in \ell^p.$ If $p = \infty,$ then $\ell^$ 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 $\, x\, _\infty ~=~ \sup_n , x_n, ,$ $\ell^$ is also a Banach space. If $0 < p < 1,$ then $\ell^p$ does not carry a norm, but rather a metric defined by $d(x,y) ~=~ \sum_n \left, x_n - y_n\^p.\,$

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

A is any sequence $x_ \in \mathbb^$ such that $\lim_ x_n$ exists. The set of all convergent sequences is a vector subspace of $\mathbb^$ called the . Since every convergent sequence is bounded, $c$ is a linear subspace of $\ell^.$ Moreover, this sequence space is a closed subspace of $\ell^$ 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 $0$ is called a and is said to . The set of all sequences that converge to $0$ is a closed vector subspace of $c$ 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 $c_0$ 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 $\left\left(x_\right\right)_$ where $x_ = 1/k$ for the first $n$ entries (for $k = 1, \ldots, n$) and is zero everywhere else (that is, $\left\left(x_\right\right)_ = \left\left(1, 1/2, \ldots, 1/\left(n-1\right), 1/n, 0, 0, \ldots\right\right)$) 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 $c_.$

## 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 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 $\mathbb^n$ 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 ...
endowed with the Euclidean topology and let $\operatorname_ : \mathbb^n \to \mathbb^$ denote the canonical inclusion :$\operatorname_\left\left(x_1, \ldots, x_n\right\right) = \left\left(x_1, \ldots, x_n, 0, 0, \ldots \right\right)$. 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-dimension ...
of each inclusion is :$\operatorname \left\left( \operatorname_ \right\right) = \left\ = \mathbb^n \times \left\$ and consequently, :$\mathbb^ = \bigcup_ \operatorname \left\left( \operatorname_ \right\right).$ This family of inclusions gives $\mathbb^$ a final topology $\tau^$, 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, locally convex,
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 that is Fréchet–Urysohn. The topology $\tau^$ 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 $\mathbb^$ 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 $\operatorname \left\left( \operatorname_ \right\right)$ and $v_ \to v$ under the natural topology of that image. Often, each image $\operatorname \left\left( \operatorname_ \right\right)$ is identified with the corresponding $\mathbb^n$; explicitly, the elements $\left\left( x_1, \ldots, x_n \right\right) \in \mathbb^n$ and $\left\left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right\right)$ are identified. This is facilitated by the fact that the subspace topology on $\operatorname \left\left( \operatorname_ \right\right)$, 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 $\operatorname_$, and the Euclidean topology on $\mathbb^n$ all coincide. With this identification, $\left\left( \left\left(\mathbb^, \tau^\right\right), \left\left(\operatorname_\right\right)_\right\right)$ 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 categ ...
of the directed system $\left\left( \left\left(\mathbb^n\right\right)_, \left\left(\operatorname_\right\right)_,\N \right\right),$ where every inclusion adds trailing zeros: :$\operatorname_\left\left(x_1, \ldots, x_m\right\right) = \left\left(x_1, \ldots, x_m, 0, \ldots, 0 \right\right)$. This shows $\left\left(\mathbb^, \tau^\right\right)$ 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, 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 ...
:$\left(x_n\right)_ \mapsto \left\left(\sum_^n x_i\right\right)_.$ 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 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, 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 de ...
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 ...
:$\, x+y\, _p^2 + \, x-y\, _p^2= 2\, x\, _p^2 + 2\, y\, _p^2.$ Substituting two distinct unit vectors for ''x'' and ''y'' directly shows that the identity is not true unless ''p'' = 2. Each is distinct, in that is a strict subset 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 i ...
to when . In fact, by Pitt's theorem , every bounded linear operator from to is compact 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. 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 the functional $L_x(y) = \sum_n x_n y_n$ for ''y'' in . Hölder's inequality implies that ''L''''x'' is a bounded linear functional on , 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 with :$y_n = \begin 0&\text\ x_n=0\\ x_n^, x_n, ^q &\text~ x_n \neq 0 \end$ gives ''L''''x''(''y'') = , , ''x'', , ''q'', so that in fact :$\, L_x\, _ = \, x\, _q.$ Conversely, given a bounded linear functional ''L'' on , the sequence defined by lies in ℓ''q''. Thus the mapping $x\mapsto L_x$ gives an isometry $\kappa_q : \ell^q \to (\ell^p)^*.$ The map :$\ell^q\xrightarrow\left(\ell^p\right)^*\xrightarrow$ 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 t ...
coincides with the canonical injection of ℓ''q'' into its double dual. 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, 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 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 basis , 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 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 top ...
, 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, 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 vec ...
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 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.

For

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

Let H be a separable Hilbert space. Every orthogonal set in H is at most countable (i.e. has finite
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
or $\,\aleph_0\,$). The following two items are related: * If H is infinite dimensional, then it is isomorphic to ''ℓ''2 * If , then H is isomorphic to $\Complex^N$

# Properties of ''ℓ''1 spaces

A sequence of elements in ''ℓ''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 $\varepsilon > 0$, there exists a natural number $n_ \geq 0$ such that $\sum_^ , s_n , < \varepsilon$ for all $s = \left\left( s_n \right\right)_^ \in K$.