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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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and related areas of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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'', can be added together and multiplied ("scaled") by numbers called sc ...
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 cal ...
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 a ...
. Equivalently, it is a
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
whose elements are functions from the
natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
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'', can be added together and multiplied ("scaled") by numbers called sc ...
under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are
linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...
s of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. 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 sequences or null sequences form sequence spaces, respectively denoted ''c'' and ''c''0, with the sup norm. Any sequence space can also be equipped with the
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
of pointwise convergence, under which it becomes a special kind of Fréchet space 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 cal ...
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 set \mathbb^ of all sequences 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'', can be added together and multiplied ("scaled") by numbers called sc ...
for componentwise addition :\left(x_n\right)_ + \left(y_n\right)_ = \left(x_n + y_n\right)_, and componentwise scalar multiplication :\alpha\left(x_n\right)_ = \left(\alpha x_n\right)_. A sequence space is any
linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...
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-seemin ...
. Under this topology, \mathbb^ is Fréchet, meaning that it is a complete, metrizable, locally convex
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(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, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...
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'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
.


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 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 with respect to this norm, and therefore is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
. If p = 2 then \ell^2 is also a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
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, ofte ...
, 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 sequences 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, 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 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(x_\right)_ where x_ = 1/k for the first n entries (for k = 1, \ldots, n) and is zero everywhere else (that is, \left(x_\right)_ = \left(1, 1/2, \ldots, 1/(n-1), 1/n, 0, 0, \ldots\right)) is a Cauchy sequence 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
let \mathbb^n denote the usual
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
endowed with the
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...
and let \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 image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
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 \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,
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
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'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative 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 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 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, 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 :(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 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, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, 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, ofte ...
should satisfy 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 is distinct, in that is a strict
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of whenever ''p'' < ''s''; furthermore, is not linearly
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
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 In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces. The numbers an ...
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(\ell^p)^*\xrightarrow(\ell^q)^ obtained by composing κ''p'' with the inverse of its
transpose In linear algebra, the transpose of a Matrix (mathematics), 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 ...
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 for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
. 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 In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
, where ''e''''i'' is the sequence which is zero but for a 1 in the ''i'' th entry. The space 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 ℓ''p'' spaces can be embedded into many
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
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.


''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 \, x\, _q\le\, x\, _p. Indeed, the inequality is homogeneous in the x_i, so it is sufficient to prove it under the assumption that \, x\, _p = 1. In this case, we need only show that \textstyle\sum , x_i, ^q \le 1 for q>p. But if \, x\, _p = 1, then , x_i, \le 1 for all i, and then \textstyle\sum , x_i, ^q \le \textstyle\sum , x_i, ^p = 1.


''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 coo ...
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( s_n \right)_^ \in K.


See also

* Lp space * Tsirelson space * beta-dual space * Orlicz sequence space *
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...


References


Bibliography

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