Sequence space

{{Short description|Vector space of infinite sequences}} {{About||usage in evolutionary biology|Sequence space (evolution)|mathematical operations on sequence numbers|Serial number arithmetic}} {{Use American English|date = March 2019}}

In functional analysis and related areas of mathematics, a '''sequence space''' is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field {{tmath|\mathbb 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 {{tmath|\mathbb K}}, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces 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 {{tmath|\textstyle \ell^p}} spaces, consisting of the {{tmath|p}}-power summable sequences, with the {{tmath|p}}-norm. These are special cases of {{tmath|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 {{tmath|c}} and {{tmath|c_0}}, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

== Definition ==

A sequence <math>\textstyle x_{\bull} = (x_n)_{n \in \N}</math> in a set {{tmath|X}} is an {{tmath|X}}-valued map <math>x_{\bull} : \N \to X</math> whose value at {{tmath|n \in \N}} is denoted by {{tmath|x_n}} instead of the usual parentheses notation {{tmath|x(n)}}.

=== Space of all sequences{{anchor|Space of all sequences}}{{anchor|Space of all real sequences}} ===

Let {{tmath|\mathbb K}} denote the field either of real or complex numbers. The set {{tmath|\textstyle \mathbb{K}^\N}} of all sequences of elements of {{tmath|\mathbb K}} is a vector space for componentwise addition <math display=block>\left(x_n\right)_{n \in \N} + \left(y_n\right)_{n \in \N} = \left(x_n + y_n\right)_{n \in \N},</math> and componentwise scalar multiplication <math display=block>\alpha\left(x_n\right)_{n \in \N} = \left(\alpha x_n\right)_{n \in \N}.</math>

A '''sequence space''' is any linear subspace of {{tmath|\textstyle \mathbb{K}^\N}}.

As a topological space, {{tmath|\textstyle \mathbb{K}^\N}} is naturally endowed with the product topology. Under this topology, {{tmath|\textstyle \mathbb{K}^\N}} is Fréchet, meaning that it is a complete, metrizable, locally convex topological vector space (TVS). However, this topology is rather pathological: there are no continuous norms on {{tmath|\textstyle \mathbb{K}^\N}} (and thus the product topology cannot be defined by any norm).{{sfn|Jarchow|1981|pp=129-130}} Among Fréchet spaces, {{tmath|\textstyle \mathbb{K}^\N}} is minimal in having no continuous norms:

{{Math theorem | name = Theorem{{sfn|Jarchow|1981|pp=129-130}} | math_statement = Let {{tmath|X}} be a Fréchet space over {{tmath|\mathbb K}}. Then the following are equivalent: <ol> <li>{{tmath|X}} admits no continuous norm (that is, any continuous seminorm on {{tmath|X}} has a nontrivial null space).</li> <li>{{tmath|X}} contains a vector subspace TVS-isomorphic to {{tmath|\textstyle \mathbb{K}^\N}}.</li> <li>{{tmath|X}} contains a complemented vector subspace TVS-isomorphic to {{tmath|\textstyle \mathbb{K}^\N}}.</li> </ol> }}But the product topology is also unavoidable: {{tmath|\textstyle \mathbb{K}^\N}} does not admit a strictly coarser Hausdorff, locally convex topology.{{sfn|Jarchow|1981|pp=129-130}} For that reason, the study of sequences begins by finding a strict linear subspace of interest, and endowing it with a topology ''different'' from the subspace topology.

=== {{math|''ℓ''''p''}} spaces=== {{See also | Lp space {{!}} {{math|Lp}} space | L-infinity {{!}} {{math|L}}-infinity }}

For {{tmath|0 \textstyle x_{\bull} = (x_n)_{n \in \N}</math> satisfying <math display=block>\sum_n |x_n|^p < \infty.</math>

If {{tmath|p \geq 1}}, then the real-valued function <math>\|\cdot\|_p</math> on {{tmath|\textstyle \ell^p}} defined by <math display=block>\|x\|_p ~=~ \Bigl(\sum_n|x_n|^p\Bigr)^{1/p} \qquad \text{ for all } x \in \ell^p</math> defines a norm on {{tmath|\textstyle \ell^p}}. In fact, {{tmath|\textstyle \ell^p}} is a complete metric space with respect to this norm, and therefore is a Banach space.

If {{tmath|1= p = 2}} then {{tmath|\textstyle \ell^2}} is also a Hilbert space when endowed with its canonical inner product, called the '''{{visible anchor|Euclidean inner product}}''', defined for all {{tmath|\textstyle x_\bull, y_\bull \in \ell^p}} by <math display=block>\langle x_\bull, y_\bull \rangle ~=~ \sum_n \overline{x_n\!}\, y_n.</math> The canonical norm induced by this inner product is the usual {{tmath|\textstyle \ell^2}}-norm, meaning that <math>\textstyle \|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}</math> for all {{tmath|\textstyle \mathbf{x} \in \ell^p}}.

If {{tmath|1= p = \infty}}, then {{tmath|\textstyle \ell^\infty}} is defined to be the space of all bounded sequences endowed with the norm <math display=block>\|x\|_\infty ~=~ \sup_n |x_n|,</math> {{tmath|\textstyle \ell^\infty}} is also a Banach space.

If {{tmath|0 d(x,y) ~=~ \sum_n \left|x_n - y_n\right|^p.</math>

==={{math|''c''}}, {{math|''c''0}} and {{math|''c''00}} === {{See also|c space}}

A {{em|convergent sequence}} is any sequence <math>\textstyle x_{\bull} \in \mathbb{K}^\N</math> such that <math>\textstyle \lim_{n \to \infty} x_n</math> exists. The set {{visible anchor|c|text={{tmath|c}} }} of all convergent sequences is a vector subspace of {{tmath|\textstyle \mathbb{K}^\N<}} called the {{em|{{visible anchor|space of convergent sequences}}}}. Since every convergent sequence is bounded, {{tmath|c}} is a linear subspace of {{tmath|\ell^\infty}}. Moreover, this sequence space is a closed subspace of {{tmath|\textstyle \ell^\infty}} with respect to the supremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to {{tmath|0}} is called a {{em|null sequence}} and is said to {{em|{{visible anchor|vanish}}}}. The set of all sequences that converge to {{tmath|0}} is a closed vector subspace of {{tmath|c}} that when endowed with the supremum norm becomes a Banach space that is denoted by {{visible anchor|c0|text={{tmath|c_0}}}} and is called the {{em|{{visible anchor|space of null sequences}}}} or the {{em|{{visible anchor|space of vanishing sequences}}}}.

The {{em|{{visible anchor|space of eventually zero sequences}}}}, {{visible anchor|c00|text={{tmath|c_{00} }}}}, is the subspace of {{tmath|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 <math>\textstyle (x_{nk})_{k \in \N}</math> where <math>x_{nk} = 1/k</math> for the first <math>n</math> entries (for <math>k = 1, \ldots, n</math>) and is zero everywhere else (that is, <math>\textstyle (x_{nk})_{k \in \N} = {}\!</math><math>\bigl(1, \tfrac12, \ldots,{}</math><math> \tfrac{1}{n-1}, \tfrac{1}{n}, {}</math><math>0, 0, \ldots\bigr)</math>) is a Cauchy sequence but it does not converge to a sequence in <math>c_{00}.</math>

=== Space of all finite sequences === {{anchor|Space of finite sequences}}

Let <math display=block>\mathbb{K}^\infty=\left\{\left(x_1, x_2,\ldots\right)\in\mathbb{K}^\N : \text{all but finitely many }x_i\text{ equal }0\right\} </math>

denote the '''space of finite sequences over''' {{tmath|\mathbb K}}. As a vector space, <math>\textstyle \mathbb{K}^\infty</math> is equal to {{tmath|c_{00} }}, but {{tmath|\textstyle \mathbb{K}^\infty}} has a different topology.

For every natural number {{tmath|n \in \N}}, let {{tmath|\textstyle \mathbb{K}^n}} denote the usual Euclidean space endowed with the Euclidean topology and let <math>\textstyle \operatorname{In}_{\mathbb{K}^n} : \mathbb{K}^n \to \mathbb{K}^\infty</math> denote the canonical inclusion <math display=block>\operatorname{In}_{\mathbb{K}^n}\left(x_1, \ldots, x_n\right) = \left(x_1, \ldots, x_n, 0, 0, \ldots \right).</math> The image of each inclusion is <math display=block> \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right) = \left\{ \left(x_1, \ldots, x_n, 0, 0, \ldots \right) : x_1, \ldots, x_n \in \mathbb{K} \right\} = \mathbb{K}^n \times \left\{ (0, 0, \ldots) \right\} </math> and consequently, <math display=block> \mathbb{K}^\infty = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right). </math>

This family of inclusions gives {{tmath|\textstyle \mathbb{K}^\infty}} a final topology {{tmath|\textstyle \tau^\infty}}, defined to be the finest topology on {{tmath|\textstyle \mathbb{K}^\infty}} such that all the inclusions are continuous (an example of a coherent topology). With this topology, {{tmath|\textstyle \mathbb{K}^\infty}} becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is {{em|not}} Fréchet–Urysohn. The topology {{tmath|\textstyle \tau^\infty}} is also strictly finer than the subspace topology induced on {{tmath|\textstyle \mathbb{K}^\infty}} by {{tmath|\textstyle \mathbb{K}^\N}}.

Convergence in {{tmath|\textstyle \tau^\infty}} has a natural description: if <math>\textstyle v \in \mathbb{K}^\infty</math> and {{tmath|v_\bull}} is a sequence in {{tmath|\textstyle \mathbb{K}^\infty}} then {{tmath|v_\bull \to v}} in {{tmath|\textstyle \tau^\infty}} if and only {{tmath|v_\bull}} is eventually contained in a single image <math>\textstyle \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right)</math> and {{tmath|v_\bull \to v}} under the natural topology of that image.

Often, each image <math>\textstyle \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right)</math> is identified with the corresponding {{tmath|\textstyle \mathbb{K}^n}}; explicitly, the elements <math>\textstyle \left( x_1, \ldots, x_n \right) \in \mathbb{K}^n</math> and <math>\left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right)</math> are identified. This is facilitated by the fact that the subspace topology on {{nobr|<math>\textstyle \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right)</math>,}} the quotient topology from the map {{nobr|<math>\textstyle \operatorname{In}_{\mathbb{K}^n}</math>,}} and the Euclidean topology on {{tmath|\textstyle \mathbb{K}^n}} all coincide. With this identification, <math>\textstyle \left( \left(\mathbb{K}^\infty, \tau^\infty\right), \left(\operatorname{In}_{\mathbb{K}^n}\right)_{n \in \N}\right)</math> is the direct limit of the directed system <math>\textstyle \left( \left(\mathbb{K}^n\right)_{n \in \N}, \left(\operatorname{In}_{\mathbb{K}^m\to\mathbb{K}^n}\right)_{m \leq n\in\N},\N \right),</math> where every inclusion adds trailing zeros: <math display=block>\operatorname{In}_{\mathbb{K}^m\to\mathbb{K}^n}\left(x_1, \ldots, x_m\right) = \left(x_1, \ldots, x_m, 0, \ldots, 0 \right).</math> This shows <math>\textstyle \left(\mathbb{K}^\infty, \tau^\infty \right)</math> is an LB-space.

=== Other sequence spaces === The space of bounded series, denote by bs, is the space of sequences {{tmath|x}} for which <math display=block>\sup_n \biggl\vert \sum_{i=0}^n x_i \biggr\vert < \infty.</math>

This space, when equipped with the norm <math display=block>\|x\|_{bs} = \sup_n \biggl\vert \sum_{i=0}^n x_i \biggr\vert,</math>

is a Banach space isometrically isomorphic to <math>\textstyle \ell^\infty,</math> via the linear mapping <math display=block>(x_n)_{n \in \N} \mapsto \biggl(\sum_{i=0}^n x_i\biggr)_{n \in \N}.</math>

The subspace <math>cs</math> consisting of all convergent series is a subspace that goes over to the space {{tmath|c}} under this isomorphism.

The space {{tmath|\Phi}} or <math>c_{00}</math> 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 {{math|''ℓ''''p''}} spaces and the space {{math|''c''0}} == {{See also|c space}} The space {{tmath|\textstyle \ell^2}} is the only {{tmath|\textstyle \ell^p}} space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law

Substituting two distinct unit vectors for {{tmath|x}} and {{tmath|y}} directly shows that the identity is not true unless {{tmath|1= p = 2}}.

Each {{tmath|\textstyle \ell^p}} is distinct, in that {{tmath|\textstyle \ell^p}} is a strict subset of {{tmath|\textstyle \ell^s}} whenever {{tmath|p < s}}; furthermore, {{tmath|\textstyle \ell^p}} is not linearly isomorphic to {{tmath|\textstyle \ell^s}} when {{tmath|p \neq s}}. In fact, by Pitt's theorem {{harv|Pitt|1936}}, every bounded linear operator from {{tmath|\textstyle \ell^s}} to {{tmath|\textstyle \ell^p}} is compact when {{tmath|p < s}}. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of {{tmath|\ell^s}}, and is thus said to be strictly singular.

If {{tmath|1 < p < \infty}}, then the (continuous) dual space of {{tmath|\textstyle \ell^p}} is isometrically isomorphic to {{tmath|\textstyle \ell^q}}, where {{tmath|q}} is the Hölder conjugate of {{tmath|p}}: {{tmath|1= 1/p + 1/q = 1}}. The specific isomorphism associates to an element {{tmath|x}} of {{tmath|\textstyle \ell^q}} the functional <math display=block>L_x(y) = \sum_n x_n y_n</math> for {{tmath|y}} in {{tmath|\textstyle \ell^p}}. Hölder's inequality implies that {{tmath|L_x}} is a bounded linear functional on {{tmath|\textstyle \ell^p}}, and in fact <math display=block> |L_x(y)| \le \|x\|_q\, \|y\|_p </math> so that the operator norm satisfies <math display=block>\|L_x\|_{(\ell^p)^*} \mathrel{\stackrel{\rm{def}}{=}} \sup_{y\in\ell^p, y\not=0} \frac{|L_x(y)|}{\|y\|_p} \le \|x\|_q.</math> In fact, taking {{tmath|y}} to be the element of {{tmath|\textstyle \ell^p}} with <math display=block> y_n = \begin{cases} 0 & \text{if}\ x_n=0 \\ x_n^{-1}|x_n|^q & \text{if}~ x_n \neq 0 \end{cases}</math> gives {{nobr|<math>L_x(y) = \|x\|_q</math>,}} so that in fact <math display=block>\|L_x\|_{(\ell^p)^*} = \|x\|_q.</math> Conversely, given a bounded linear functional {{tmath|L}} on {{tmath|\textstyle \ell^p}}, the sequence defined by {{tmath|1= x_n = L(e_n)}} lies in {{tmath|\textstyle \ell^q}}. Thus the mapping {{tmath|x\mapsto L_x}} gives an isometry <math display=block> \kappa_q : \ell^q \to (\ell^p)^*. </math>

The map <math display=block>\ell^q\xrightarrow{\kappa_q}(\ell^p)^*\xrightarrow{(\kappa_q^*)^{-1}}(\ell^q)^{**}</math> obtained by composing {{tmath|\kappa_p}} with the inverse of its transpose coincides with the canonical injection of {{tmath|\textstyle \ell^q}} into its double dual. As a consequence {{tmath|\textstyle \ell^q}} is a reflexive space. By abuse of notation, it is typical to identify {{tmath|\textstyle \ell^q}} with the dual of {{tmath|\textstyle \ell^p}}: {{tmath|1=\textstyle (\ell^p)^* = \ell^q }}. Then reflexivity is understood by the sequence of identifications {{tmath|1=\textstyle (\ell^p)^{**} = (\ell^q)^* = \ell^p}}.

The space {{tmath| c_0}} is defined as the space of all sequences converging to zero, with norm identical to {{nobr|<math>\|x\|_\infty</math>.}} It is a closed subspace of {{tmath|\textstyle \ell^\infty}}, hence a Banach space. The dual of {{tmath|c_0}} is {{tmath|\textstyle \ell^1}}; the dual of {{tmath|\textstyle \ell^1}} is {{tmath|\textstyle \ell^\infty}}. For the case of natural numbers index set, the {{tmath|\textstyle \ell^p}} and {{tmath|c_0}} are separable, with the sole exception of {{tmath|\textstyle \ell^\infty}}. The dual of {{tmath|\textstyle \ell^\infty}} is the ba space.

The spaces {{tmath|c_0}} and {{tmath|\textstyle \ell^p}} (for {{tmath|1 \leq p < \infty}}) have a canonical unconditional Schauder basis {{tmath|1= \{e_i : i = 1, 2, \ldots \} }}, where {{tmath|e_i}} is the sequence which is zero but for a {{tmath|1}} in the {{tmath|i}}th entry.

The space ℓ1 has the Schur property: In ℓ1, any sequence that is weakly convergent is also strongly convergent {{harv|Schur|1921}}. 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 {{tmath|\textstyle \ell^p}} spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some {{tmath|\textstyle \ell^p}} or of {{tmath|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 {{tmath|\textstyle \ell^1}}, was answered in the affirmative by {{harvtxt|Banach|Mazur|1933}}. That is, for every separable Banach space {{tmath|X}}, there exists a quotient map {{tmath|\textstyle Q: \ell^1 \to X}}, so that {{tmath|X}} is isomorphic to {{tmath|\textstyle \ell^1 / \ker Q}}. In general, {{tmath| \operatorname{ker} Q}} is not complemented in {{tmath|\textstyle \ell^1}}, that is, there does not exist a subspace {{tmath|Y}} of {{tmath|\textstyle \ell^1}} such that {{tmath|1=\textstyle \ell^1 = Y \oplus \ker Q }}. In fact, {{tmath|\textstyle \ell^1}} has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take {{tmath|1=\textstyle X=\ell^p}}; since there are uncountably many such {{tmath|X}}{{'}}s, and since no {{tmath|\textstyle \ell^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 {{tmath|\textstyle \ell^q}} is that it is not polynomially reflexive.

=== {{math|''ℓ''''p''}} spaces are increasing in {{mvar|p}} === For {{tmath| p \in [1,\infty] }}, the spaces {{tmath|\textstyle \ell^p}} are increasing in {{tmath|p}}, with the inclusion operator being continuous: for {{tmath|1 \le p < q \le \infty}}, one has {{nobr|<math>\|x\|_q\le\|x\|_p</math>.}} Indeed, the inequality is homogeneous in the {{tmath|x_i}}, so it is sufficient to prove it under the assumption that {{nobr|<math>\|x\|_p = 1</math>.}} In this case, we need only show that <math>\textstyle\sum |x_i|^q \le 1</math> for {{tmath|q > p}}. But if {{nobr|<math>\|x\|_p = 1</math>,}} then <math>|x_i|\le 1</math> for all {{tmath|i}}, and then <math>\textstyle \sum |x_i|^q \le {}\!</math>{{nobr|<math>\textstyle\sum |x_i|^p = 1</math>.}}

=== {{math|''ℓ''2}} is isomorphic to all separable, infinite dimensional Hilbert spaces === Let {{tmath|H}} be a separable Hilbert space. Every orthogonal set in {{tmath|H}} is at most countable (i.e. has finite dimension or {{tmath|\aleph_0}}).<ref name="Debnath, Mikusinski-2005">{{cite book | last1 = Debnath | first1 = Lokenath | last2 = Mikusinski | first2 = Piotr | title=Hilbert Spaces with Applications | publisher=Elsevier | isbn= 978-0-12-2084386 | pages=120–121 | year=2005}}</ref> The following two items are related: * If {{tmath|H}} is infinite dimensional, then it is isomorphic to {{tmath|\textstyle \ell^2}}, * If {{tmath|1= \operatorname{dim}(H) = N}}, then {{tmath|H}} is isomorphic to {{tmath|\textstyle \C^N}}.

== Properties of {{math|''ℓ''1}} spaces == A sequence of elements in {{tmath|\textstyle \ell^1}} converges in the space of complex sequences {{tmath|\textstyle \ell^1}} if and only if it converges weakly in this space.{{sfn | Trèves | 2006 | pp=451-458}} If {{tmath|K}} is a subset of this space, then the following are equivalent:{{sfn | Trèves | 2006 | pp=451-458}} # {{tmath|K}} is compact; # {{tmath|K}} is weakly compact; # {{tmath|K}} is bounded, closed, and equismall at infinity. Here {{tmath|K}} being '''equismall at infinity''' means that for every {{tmath|\varepsilon > 0}}, there exists a natural number <math>n_{\varepsilon} \geq 0</math> such that <math>\textstyle \sum_{n = n_{\epsilon}}^\infty | s_n | < \varepsilon</math> for all {{tmath|1=\textstyle s = \left( s_n \right)_{n=1}^\infty \in K}}.

== References == {{Reflist}}

==Bibliography== * {{citation | last1=Banach | first1=Stefan | last2=Mazur | first2=S. | title=Zur Theorie der linearen Dimension | journal=Studia Mathematica | volume=4 | year=1933 | pages=100–112| doi=10.4064/sm-4-1-100-112 }}. * {{citation | last1=Dunford | first1=Nelson| last2=Schwartz | first2=Jacob T. | title=Linear operators, volume I | publisher=Wiley-Interscience | year=1958}}. * {{Jarchow Locally Convex Spaces}}  * {{citation | last=Pitt | first=H.R. | doi=10.1112/jlms/s1-11.3.174 | title=A note on bilinear forms | journal=J. London Math. Soc. | volume=11 | issue=3 | year=1936 | pages=174–180}}. * {{Narici Beckenstein Topological Vector Spaces|edition=2}}  * {{Schaefer Wolff Topological Vector Spaces|edition=2}}  * {{citation | last=Schur | first=J. | title=Über lineare Transformationen in der Theorie der unendlichen Reihen | journal=Journal für die reine und angewandte Mathematik|volume=151|year=1921|pages=79–111|doi=10.1515/crll.1921.151.79}}. * {{Trèves François Topological vector spaces, distributions and kernels}}

Category:Sequence spaces Category:Functional analysis Category:Sequences and series