View source for Sequence space

{{Short description|Vector space of infinite sequences}}

In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], a '''sequence space''' is a [[Vector space|vector space]] whose elements are infinite [[Sequence|sequence]]s of [[Real number|real]] or complex numbers. Equivalently, it is a [[Function space|function space]] whose elements are functions from the natural numbers to the [[Field (mathematics)|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|vector space]] under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are [[Linear subspace|linear subspace]]s of this space.  Sequence spaces are typically equipped with a [[Norm (mathematics)|norm]], or at least the structure of a [[Topological vector space|topological vector space]].

The most important sequence spaces in analysis are the {{math|''ℓ''{{i sup|''p''}}}} spaces, consisting of the {{math|''p''}}-power summable sequences, with the ''p''-norm.  These are special cases of [[Lp space|L<sup>''p''</sup> spaces]] for the [[Counting measure|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''<sub>0</sub>, with the sup norm. Any sequence space can also be equipped with the [[Topology|topology]] of [[Pointwise convergence|pointwise convergence]], under which it becomes a special kind of [[Fréchet space]] called [[FK-space]].

== Definition ==

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

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

Let <math>\mathbb{K}</math> denote the field either of real or complex numbers. The set <math>\mathbb{K}^{\N}</math> of all sequences of elements of <math>\mathbb{K}</math> is a [[Vector space|vector space]] for componentwise addition
:<math>\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|scalar multiplication]]
:<math>\alpha\left(x_n\right)_{n \in \N} = \left(\alpha x_n\right)_{n \in \N}.</math>

A '''sequence space''' is any [[Linear subspace|linear subspace]] of <math>\mathbb{K}^{\N}.</math> 

As a topological space, <math>\mathbb{K}^{\N}</math> is naturally endowed with the [[Product topology|product topology]].  Under this topology, <math>\mathbb{K}^{\N}</math> is [[Fréchet space|Fréchet]], meaning that it is a [[Complete topological vector space|complete]], [[Metrizable topological vector space|metrizable]], [[Locally convex topological vector space|locally&nbsp;convex]] [[Topological vector space|topological vector space]] (TVS).  However, this topology is rather pathological: there are no [[Continuous function|continuous]] norms on <math>\mathbb{K}^{\N}</math> (and thus the product topology cannot be defined by any [[Norm (mathematics)|norm]]).{{sfn|Jarchow|1981|pp=129-130}}  Among Fréchet spaces, <math>\mathbb{K}^{\N}</math> is minimal in having no continuous norms: 

{{Math theorem
| name = Theorem{{sfn|Jarchow|1981|pp=129-130}}
| math_statement = Let <math>X</math> be a [[Fréchet space]] over <math>\mathbb{K}.</math> 
Then the following are equivalent:
<ol>
<li><math>X</math> admits no continuous norm (that is, any continuous seminorm on <math>X</math> has a nontrivial null space).</li>
<li><math>X</math> contains a vector subspace TVS-isomorphic to <math>\mathbb{K}^{\mathbb{N} }</math>.</li>
<li><math>X</math> contains a complemented vector subspace TVS-isomorphic to <math>\mathbb{K}^{\mathbb{N} }</math>.</li>
</ol>
}}But the product topology is also unavoidable: <math>\mathbb{K}^{\N}</math> does not admit a [[Comparison of topologies|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|linear subspace]] of interest, and endowing it with a topology ''different'' from the [[Subspace topology|subspace topology]].

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

For <math>0 < p < \infty,</math> <math>\ell^p</math> is the subspace of <math>\mathbb{K}^{\N}</math> consisting of all sequences <math>x_{\bull} = \left(x_n\right)_{n \in \N}</math> satisfying
<math display=block>\sum_n |x_n|^p < \infty.</math>

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

If <math>p = 2</math> then <math>\ell^2</math> is also a [[Hilbert space]] when endowed with its canonical inner product, called the '''{{visible anchor|Euclidean inner product}}''', defined for all <math>x_\bull, y_\bull \in \ell^p</math> 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 <math>\ell^2</math>-norm, meaning that <math>\|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}</math> for all <math>\mathbf{x} \in \ell^p.</math> 

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

If <math>0 < p < 1,</math> then <math>\ell^p</math> does not carry a norm, but rather a [[Metric space|metric]] defined by
<math display=block>d(x,y) ~=~ \sum_n \left|x_n - y_n\right|^p.\,</math>

===''c'', ''c''<sub>0</sub> and  ''c''<sub>00</sub>===

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

A sequence that converges to <math>0</math> is called a {{em|null sequence}} and is said to {{em|{{visible anchor|vanish}}}}. The set of all sequences that converge to <math>0</math> is a closed vector subspace of <math>c</math> that when endowed with the supremum norm becomes a Banach space that is denoted by {{visible anchor|c0|text=<math>c_0</math>}} 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=<math>c_{00},</math>}} is the subspace of <math>c_0</math> 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>\left(x_{nk}\right)_{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>\left(x_{nk}\right)_{k \in \N} = \left(1, 1/2, \ldots, 1/(n-1), 1/n, 0, 0, \ldots\right)</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>\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''' <math>\mathbb{K}</math>.  As a vector space, <math>\mathbb{K}^{\infty}</math> is equal to <math>c_{00}</math>, but <math>\mathbb{K}^{\infty}</math> has a different topology.  

For every [[Natural number|natural number]] {{nowrap|<math>n \in \N</math>,}} let <math>\mathbb{K}^n</math> denote the usual [[Euclidean space]] endowed with the [[Euclidean topology]] and let <math>\operatorname{In}_{\mathbb{K}^n} : \mathbb{K}^n \to \mathbb{K}^{\infty}</math> denote the canonical inclusion 
:<math>\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 (mathematics)|image]] of each inclusion is
:<math>\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>\mathbb{K}^{\infty} = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right).</math> 

This family of inclusions gives <math>\mathbb{K}^{\infty}</math> a [[Final topology|final topology]] <math>\tau^{\infty}</math>, defined to be the [[Comparison of topologies|finest topology]] on <math>\mathbb{K}^{\infty}</math> such that all the inclusions are continuous (an example of a [[Coherent topology|coherent topology]]).  With this topology, <math>\mathbb{K}^{\infty}</math> becomes a [[Complete topological vector space|complete]], [[Hausdorff space|Hausdorff]], [[Locally convex topological vector space|locally convex]], [[Sequential space|sequential]], [[Topological vector space|topological vector space]] that is {{em|not}} [[Fréchet–Urysohn space|Fréchet–Urysohn]].  The topology <math>\tau^{\infty}</math> is also [[Comparison of topologies|strictly finer]] than the [[Subspace topology|subspace topology]] induced on <math>\mathbb{K}^{\infty}</math> by <math>\mathbb{K}^{\N}</math>.   

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

Often, each image <math>\operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right)</math> is identified with the corresponding <math>\mathbb{K}^n</math>; explicitly, the elements <math>\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 <math>\operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right)</math>, the quotient topology from the map <math>\operatorname{In}_{\mathbb{K}^n}</math>, and the Euclidean topology on <math>\mathbb{K}^n</math> all coincide.  With this identification, <math>\left( \left(\mathbb{K}^{\infty}, \tau^{\infty}\right), \left(\operatorname{In}_{\mathbb{K}^n}\right)_{n \in \N}\right)</math> is the [[Direct limit|direct limit]] of the directed system <math>\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>\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>\left(\mathbb{K}^{\infty}, \tau^{\infty}\right)</math> is an [[LB-space]].

=== Other sequence spaces ===
The space of bounded [[Series (mathematics)|series]], denote by [[Bs space|bs]], is the space of sequences <math>x</math> for which
:<math>\sup_n \left\vert \sum_{i=0}^n x_i \right\vert < \infty.</math>

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

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

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 <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 set|dense]] in many sequence spaces.

== Properties of ℓ<sup>''p''</sup> spaces and the space ''c''<sub>0</sub> ==
The space ℓ<sup>2</sup> is the only ℓ<sup>''p''</sup> space that is a [[Hilbert space]], since any norm that is induced by an inner product should satisfy the [[Parallelogram law|parallelogram law]]

:<math>\|x+y\|_p^2 + \|x-y\|_p^2= 2\|x\|_p^2 + 2\|y\|_p^2.</math>

Substituting two distinct unit vectors for ''x'' and ''y'' directly shows that the identity is not true unless ''p''&thinsp;=&thinsp;2.

Each {{math|''ℓ''<sup>''p''</sup>}} is distinct, in that {{math|''ℓ''<sup>''p''</sup>}} is a strict [[Subset|subset]] of {{math|''ℓ''<sup>''s''</sup>}} whenever ''p''&thinsp;<&thinsp;''s''; furthermore, {{math|''ℓ''<sup>''p''</sup>}} is not linearly isomorphic to {{math|''ℓ''<sup>''s''</sup>}} when&nbsp;{{math|''p'' ≠ ''s''}}.  In fact, by Pitt's theorem {{harv|Pitt|1936}}, every bounded linear operator from {{math|''ℓ''<sup>''s''</sup>}} to {{math|''ℓ''<sup>''p''</sup>}} is [[Compact operator|compact]] when {{math|''p'' < ''s''}}.  No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of {{math|''ℓ''<sup>''s''</sup>}}, and is thus said to be strictly singular.

If 1&thinsp;<&thinsp;''p''&thinsp;<&thinsp;∞, then the [[Dual space|(continuous) dual space]] of ℓ<sup>''p''</sup> is isometrically isomorphic to ℓ<sup>''q''</sup>, where ''q'' is the Hölder conjugate of ''p'': 1/''p''&nbsp;+ 1/''q''&nbsp;= 1.  The specific isomorphism associates to an element ''x'' of {{math|''ℓ''<sup>''q''</sup>}} the functional
<math display="block">L_x(y) = \sum_n x_n y_n</math>
for ''y'' in {{math|''ℓ''<sup>''p''</sup>}}.  [[Hölder's inequality]] implies that ''L''<sub>''x''</sub> is a bounded linear functional on {{math|''ℓ''<sup>''p''</sup>}}, and in fact
<math display="block">|L_x(y)| \le \|x\|_q\,\|y\|_p</math>
so that the operator norm satisfies
:<math>\|L_x\|_{(\ell^p)^*} \stackrel{\rm{def}}{=}\sup_{y\in\ell^p, y\not=0} \frac{|L_x(y)|}{\|y\|_p} \le \|x\|_q.</math>
In fact, taking ''y'' to be the element of {{math|''ℓ''<sup>''p''</sup>}} with
:<math>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 ''L''<sub>''x''</sub>(''y'') = ||''x''||<sub>''q''</sub>, so that in fact
:<math>\|L_x\|_{(\ell^p)^*} = \|x\|_q.</math>
Conversely, given a bounded linear functional ''L'' on {{math|''ℓ''<sup>''p''</sup>}}, the sequence defined by {{math|1=''x''<sub>''n''</sub> = ''L''(''e''<sub>''n''</sub>)}} lies in ℓ<sup>''q''</sup>.  Thus the mapping <math>x\mapsto L_x</math> gives an isometry
<math display="block">\kappa_q : \ell^q \to (\ell^p)^*.</math>

The map
:<math>\ell^q\xrightarrow{\kappa_q}(\ell^p)^*\xrightarrow{(\kappa_q^*)^{-1}}</math>
obtained by composing κ<sub>''p''</sub> with the inverse of its [[Dual space#Transpose of a continuous linear map|transpose]] coincides with the [[Reflexive space#Definitions|canonical injection]] of ℓ<sup>''q''</sup> into its double dual.  As a consequence ℓ<sup>''q''</sup> is a [[Reflexive space|reflexive space]]. By [[Abuse of notation|abuse of notation]], it is typical to identify ℓ<sup>''q''</sup> with the dual of ℓ<sup>''p''</sup>: (ℓ<sup>''p''</sup>)<sup>*</sup>&nbsp;= ℓ<sup>''q''</sup>.  Then reflexivity is understood by the sequence of identifications (ℓ<sup>''p''</sup>)<sup>**</sup>&nbsp;= (ℓ<sup>''q''</sup>)<sup>*</sup>&nbsp;= ℓ<sup>''p''</sup>.

The space ''c''<sub>0</sub> is defined as the space of all sequences converging to zero, with norm identical to  ||''x''||<sub>∞</sub>. It is a closed subspace of ℓ<sup>∞</sup>, hence a Banach space. The [[Dual space|dual]] of ''c''<sub>0</sub> is ℓ<sup>1</sup>; the dual of ℓ<sup>1</sup> is ℓ<sup>∞</sup>. For the case of natural numbers index set, the ℓ<sup>''p''</sup> and ''c''<sub>0</sub> are [[Separable space|separable]], with the sole exception of ℓ<sup>∞</sup>.  The dual of ℓ<sup>∞</sup> is the [[Ba space|ba space]].

The spaces ''c''<sub>0</sub> and ℓ<sup>''p''</sup> (for 1&nbsp;≤ ''p''&nbsp;< ∞) have a canonical unconditional [[Schauder basis]] {''e''<sub>''i''</sub>&nbsp;| ''i''&nbsp;= 1, 2,...}, where ''e''<sub>''i''</sub> is the sequence which is zero but for a 1 in the ''i''<sup>&thinsp;th</sup> entry.

The space ℓ<sup>1</sup> has the [[Schur's property|Schur property]]: In ℓ<sup>1</sup>, any sequence that is [[Weak convergence (Hilbert space)|weakly convergent]] is also [[Weak convergence (Hilbert space)|strongly convergent]] {{harv|Schur|1921}}.  However, since the [[Weak topology|weak topology]] on infinite-dimensional spaces is strictly weaker than the [[Strong topology|strong topology]], there are [[Net (mathematics)|nets]] in ℓ<sup>1</sup> that are weak convergent but not strong convergent.

The ℓ<sup>''p''</sup> spaces can be [[Embedding|embedded]] into many [[Banach space]]s. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ<sup>''p''</sup> or of ''c''<sub>0</sub>, was answered negatively by [[Biography:Boris Tsirelson|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 (linear algebra)|quotient space]] of ℓ<sup>1</sup>, was answered in the affirmative by {{harvtxt|Banach|Mazur|1933}}. That is, for every separable Banach space ''X'', there exists a quotient map <math>Q:\ell^1 \to X</math>, so that ''X'' is isomorphic to <math>\ell^1 / \ker Q</math>.  In general, ker ''Q'' is not complemented in ℓ<sup>1</sup>, that is, there does not exist a subspace ''Y'' of ℓ<sup>1</sup> such that <math>\ell^1 = Y \oplus \ker Q</math>. In fact, ℓ<sup>1</sup> has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take <math>X=\ell^p</math>; since there are uncountably many such ''X''{{'}}s, and since no ℓ<sup>''p''</sup> 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<sup>''p''</sup> is that it is not [[Polynomially reflexive space|polynomially reflexive]].

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

=== ''ℓ''<sup>2</sup> is isomorphic to all separable, infinite dimensional Hilbert spaces ===
Let H be a [[Hilbert space#Separable spaces|separable Hilbert space]].  Every orthogonal set in H is at most countable (i.e. has finite [[Hilbert space#Hilbert dimension|dimension]] or <math>\,\aleph_0\,</math>).<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 H is infinite dimensional, then it is isomorphic to ''ℓ''<sup>2</sup>
* If {{math|1=dim(''H'') = ''N''}}, then H is isomorphic to <math>\Complex^N</math>

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

== See also ==

*[[Lp space|L<sup>p</sup> space]]
*[[Tsirelson space]]
*[[Beta-dual space|beta-dual space]]
*[[Orlicz sequence space]]
*[[Hilbert space]]

== 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}}.
* {{citation | last1=Dunford | first1=Nelson| last2=Schwartz | first2=Jacob T. | title=Linear operators, volume I | publisher=Wiley-Interscience | year=1958}}.
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{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&ndash;180}}.
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{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}} <!-- {{sfn | Trèves | 2006 | p=}} -->

{{Functional analysis}}

[[Category:Sequence spaces| ]]
[[Category:Functional analysis]]
[[Category:Sequences and series]]

{{Sourceattribution|Sequence space}}