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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 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 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 ℓ spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of L 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 c₀, 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 ${\displaystyle x_{\bullet }=\left(x_{n}\right)_{n\in \mathbb {N} }}$ in a set ${\displaystyle X}$ is just an ${\displaystyle X}$-valued map ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$ whose value at ${\displaystyle n\in \mathbb {N} }$ is denoted by ${\displaystyle x_{n}}$ instead of the usual parentheses notation ${\displaystyle x(n).}$

Space of all sequences

Let ${\displaystyle \mathbb {K} }$ denote the field either of real or complex numbers. The set ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ of all sequences of elements of ${\displaystyle \mathbb {K} }$ is a vector space for componentwise addition

${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }+\left(y_{n}\right)_{n\in \mathbb {N} }=\left(x_{n}+y_{n}\right)_{n\in \mathbb {N} },}$

and componentwise scalar multiplication

${\displaystyle \alpha \left(x_{n}\right)_{n\in \mathbb {N} }=\left(\alpha x_{n}\right)_{n\in \mathbb {N} }.}$

A sequence space is any linear subspace of ${\displaystyle \mathbb {K} ^{\mathbb {N} }.}$

As a topological space, ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ is naturally endowed with the product topology. Under this topology, ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ 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 ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ (and thus the product topology cannot be defined by any norm). Among Fréchet spaces, ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ is minimal in having no continuous norms:

Theorem — Let ${\displaystyle X}$ be a Fréchet space over ${\displaystyle \mathbb {K} .}$ Then the following are equivalent:

1. ${\displaystyle X}$ admits no continuous norm (that is, any continuous seminorm on ${\displaystyle X}$ has a nontrivial null space).
2. ${\displaystyle X}$ contains a vector subspace TVS-isomorphic to ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$.
3. ${\displaystyle X}$ contains a complemented vector subspace TVS-isomorphic to ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$.

But the product topology is also unavoidable: ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ does not admit a strictly coarser Hausdorff, locally convex topology. 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.

ℓ spaces

For ${\displaystyle 0<p<\infty ,}$ ${\displaystyle \ell ^{p}}$ is the subspace of ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ consisting of all sequences ${\displaystyle x_{\bullet }=\left(x_{n}\right)_{n\in \mathbb {N} }}$ satisfying $${\displaystyle \sum _{n}|x_{n}|^{p}<\infty .}$$

If ${\displaystyle p\geq 1,}$ then the real-valued function ${\displaystyle \|\cdot \|_{p}}$ on ${\displaystyle \ell ^{p}}$ defined by $${\displaystyle \|x\|_{p}~=~\left(\sum _{n}|x_{n}|^{p}\right)^{1/p}\qquad {\text{ for all }}x\in \ell ^{p}}$$ defines a norm on ${\displaystyle \ell ^{p}.}$ In fact, ${\displaystyle \ell ^{p}}$ is a complete metric space with respect to this norm, and therefore is a Banach space.

If ${\displaystyle p=2}$ then ${\displaystyle \ell ^{2}}$ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all ${\displaystyle x_{\bullet },y_{\bullet }\in \ell ^{p}}$ by $${\displaystyle \langle x_{\bullet },y_{\bullet }\rangle ~=~\sum _{n}{\overline {x_{n}}}y_{n}.}$$ The canonical norm induced by this inner product is the usual ${\displaystyle \ell ^{2}}$-norm, meaning that ${\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}$ for all ${\displaystyle \mathbf {x} \in \ell ^{p}.}$

If ${\displaystyle p=\infty ,}$ then ${\displaystyle \ell ^{\infty }}$ is defined to be the space of all bounded sequences endowed with the norm $${\displaystyle \|x\|_{\infty }~=~\sup _{n}|x_{n}|,}$$ ${\displaystyle \ell ^{\infty }}$ is also a Banach space.

If ${\displaystyle 0<p<1,}$ then ${\displaystyle \ell ^{p}}$ does not carry a norm, but rather a metric defined by $${\displaystyle d(x,y)~=~\sum _{n}\left|x_{n}-y_{n}\right|^{p}.\,}$$

c, c₀ and c₀₀

A convergent sequence is any sequence ${\displaystyle x_{\bullet }\in \mathbb {K} ^{\mathbb {N} }}$ such that ${\displaystyle \lim _{n\to \infty }x_{n}}$ exists. The set ${\displaystyle c}$ of all convergent sequences is a vector subspace of ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ called the space of convergent sequences. Since every convergent sequence is bounded, ${\displaystyle c}$ is a linear subspace of ${\displaystyle \ell ^{\infty }.}$ Moreover, this sequence space is a closed subspace of ${\displaystyle \ell ^{\infty }}$ with respect to the supremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to ${\displaystyle 0}$ is called a null sequence and is said to vanish. The set of all sequences that converge to ${\displaystyle 0}$ is a closed vector subspace of ${\displaystyle c}$ that when endowed with the supremum norm becomes a Banach space that is denoted by ${\displaystyle c_{0}}$ and is called the space of null sequences or the space of vanishing sequences.

The space of eventually zero sequences, ${\displaystyle c_{00},}$ is the subspace of ${\displaystyle c_{0}}$ 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 ${\displaystyle \left(x_{nk}\right)_{k\in \mathbb {N} }}$ where ${\displaystyle x_{nk}=1/k}$ for the first ${\displaystyle n}$ entries (for ${\displaystyle k=1,\ldots ,n}$) and is zero everywhere else (that is, ${\displaystyle \left(x_{nk}\right)_{k\in \mathbb {N} }=\left(1,1/2,\ldots ,1/(n-1),1/n,0,0,\ldots \right)}$) is a Cauchy sequence but it does not converge to a sequence in ${\displaystyle c_{00}.}$

Space of all finite sequences

Let

${\displaystyle \mathbb {K} ^{\infty }=\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {K} ^{\mathbb {N} }:{\text{all but finitely many }}x_{i}{\text{ equal }}0\right\}}$,

denote the space of finite sequences over ${\displaystyle \mathbb {K} }$. As a vector space, ${\displaystyle \mathbb {K} ^{\infty }}$ is equal to ${\displaystyle c_{00}}$, but ${\displaystyle \mathbb {K} ^{\infty }}$ has a different topology.

For every natural number ${\displaystyle n\in \mathbb {N} }$, let ${\displaystyle \mathbb {K} ^{n}}$ denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {K} ^{n}}:\mathbb {K} ^{n}\to \mathbb {K} ^{\infty }}$ denote the canonical inclusion

${\displaystyle \operatorname {In} _{\mathbb {K} ^{n}}\left(x_{1},\ldots ,x_{n}\right)=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}$.

The image of each inclusion is

${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right)=\left\{\left(x_{1},\ldots ,x_{n},0,0,\ldots \right):x_{1},\ldots ,x_{n}\in \mathbb {K} \right\}=\mathbb {K} ^{n}\times \left\{(0,0,\ldots )\right\}}$

and consequently,

${\displaystyle \mathbb {K} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right).}$

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

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

Often, each image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right)}$ is identified with the corresponding ${\displaystyle \mathbb {K} ^{n}}$; explicitly, the elements ${\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {K} ^{n}}$ and ${\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}$ are identified. This is facilitated by the fact that the subspace topology on ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right)}$, the quotient topology from the map ${\displaystyle \operatorname {In} _{\mathbb {K} ^{n}}}$, and the Euclidean topology on ${\displaystyle \mathbb {K} ^{n}}$ all coincide. With this identification, ${\displaystyle \left(\left(\mathbb {K} ^{\infty },\tau ^{\infty }\right),\left(\operatorname {In} _{\mathbb {K} ^{n}}\right)_{n\in \mathbb {N} }\right)}$ is the direct limit of the directed system ${\displaystyle \left(\left(\mathbb {K} ^{n}\right)_{n\in \mathbb {N} },\left(\operatorname {In} _{\mathbb {K} ^{m}\to \mathbb {K} ^{n}}\right)_{m\leq n\in \mathbb {N} },\mathbb {N} \right),}$ where every inclusion adds trailing zeros:

${\displaystyle \operatorname {In} _{\mathbb {K} ^{m}\to \mathbb {K} ^{n}}\left(x_{1},\ldots ,x_{m}\right)=\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right)}$.

This shows ${\displaystyle \left(\mathbb {K} ^{\infty },\tau ^{\infty }\right)}$ is an LB-space.

Other sequence spaces

The space of bounded series, denote by bs, is the space of sequences ${\displaystyle x}$ for which

${\displaystyle \sup _{n}\left\vert \sum _{i=0}^{n}x_{i}\right\vert <\infty .}$

This space, when equipped with the norm

${\displaystyle \|x\|_{bs}=\sup _{n}\left\vert \sum _{i=0}^{n}x_{i}\right\vert ,}$

is a Banach space isometrically isomorphic to ${\displaystyle \ell ^{\infty },}$ via the linear mapping

${\displaystyle (x_{n})_{n\in \mathbb {N} }\mapsto \left(\sum _{i=0}^{n}x_{i}\right)_{n\in \mathbb {N} }.}$

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 ${\displaystyle c_{00}}$ 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 spaces and the space c₀

The space ℓ is the only ℓ space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law

${\displaystyle \|x+y\|_{p}^{2}+\|x-y\|_{p}^{2}=2\|x\|_{p}^{2}+2\|y\|_{p}^{2}.}$

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

Each ℓ is distinct, in that ℓ is a strict subset of ℓ whenever p < s; furthermore, ℓ is not linearly isomorphic to ℓ when p ≠ s. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from ℓ to ℓ is compact when p < s. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓ, and is thus said to be strictly singular.

If 1 < p < ∞, then the (continuous) dual space of ℓ is isometrically isomorphic to ℓ, where q is the Hölder conjugate of p: 1/p + 1/q = 1. The specific isomorphism associates to an element x of ℓ the functional $${\displaystyle L_{x}(y)=\sum _{n}x_{n}y_{n}}$$ for y in ℓ. Hölder's inequality implies that L_x is a bounded linear functional on ℓ, and in fact $${\displaystyle |L_{x}(y)|\leq \|x\|_{q}\,\|y\|_{p}}$$ so that the operator norm satisfies

${\displaystyle \|L_{x}\|_{(\ell ^{p})^{*}}{\stackrel {\rm {def}}{=}}\sup _{y\in \ell ^{p},y\not =0}{\frac {|L_{x}(y)|}{\|y\|_{p}}}\leq \|x\|_{q}.}$

In fact, taking y to be the element of ℓ with

${\displaystyle y_{n}={\begin{cases}0&{\text{if}}\ x_{n}=0\\x_{n}^{-1}|x_{n}|^{q}&{\text{if}}~x_{n}\neq 0\end{cases}}}$

gives L_x(y) = ||x||_q, so that in fact

${\displaystyle \|L_{x}\|_{(\ell ^{p})^{*}}=\|x\|_{q}.}$

Conversely, given a bounded linear functional L on ℓ, the sequence defined by x_n = L(e_n) lies in ℓ. Thus the mapping ${\displaystyle x\mapsto L_{x}}$ gives an isometry $${\displaystyle \kappa _{q}:\ell ^{q}\to (\ell ^{p})^{*}.}$$

The map

${\displaystyle \ell ^{q}\xrightarrow {\kappa _{q}} (\ell ^{p})^{*}\xrightarrow {(\kappa _{q}^{*})^{-1}} (\ell ^{q})^{**}}$

obtained by composing κ_p with the inverse of its transpose coincides with the canonical injection of ℓ into its double dual. As a consequence ℓ is a reflexive space. By abuse of notation, it is typical to identify ℓ with the dual of ℓ: (ℓ) = ℓ. Then reflexivity is understood by the sequence of identifications (ℓ) = (ℓ) = ℓ.

The space c₀ 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 c₀ is ℓ; the dual of ℓ is ℓ. For the case of natural numbers index set, the ℓ and c₀ are separable, with the sole exception of ℓ. The dual of ℓ is the ba space.

The spaces c₀ and ℓ (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis {e_i | i = 1, 2,...}, where e_i is the sequence which is zero but for a 1 in the i entry.

The space ℓ has the Schur property: In ℓ, any sequence that is weakly convergent is also strongly convergent (Schur 1921). However, since the weak topology on infinite-dimensional spaces is strictly weaker than the strong topology, there are nets in ℓ that are weak convergent but not strong convergent.

The ℓ spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ or of c₀, 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 ℓ, was answered in the affirmative by Banach & Mazur (1933). That is, for every separable Banach space X, there exists a quotient map ${\displaystyle Q:\ell ^{1}\to X}$, so that X is isomorphic to ${\displaystyle \ell ^{1}/\ker Q}$. In general, ker Q is not complemented in ℓ, that is, there does not exist a subspace Y of ℓ such that ${\displaystyle \ell ^{1}=Y\oplus \ker Q}$. In fact, ℓ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ${\displaystyle X=\ell ^{p}}$; since there are uncountably many such X's, and since no ℓ 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 is that it is not polynomially reflexive.

ℓ spaces are increasing in p

For ${\displaystyle p\in }$, the spaces ${\displaystyle \ell ^{p}}$ are increasing in ${\displaystyle p}$, with the inclusion operator being continuous: for ${\displaystyle 1\leq p<q\leq \infty }$, one has ${\displaystyle \|x\|_{q}\leq \|x\|_{p}}$. Indeed, the inequality is homogeneous in the ${\displaystyle x_{i}}$, so it is sufficient to prove it under the assumption that ${\displaystyle \|x\|_{p}=1}$. In this case, we need only show that ${\displaystyle \textstyle \sum |x_{i}|^{q}\leq 1}$ for ${\displaystyle q>p}$. But if ${\displaystyle \|x\|_{p}=1}$, then ${\displaystyle |x_{i}|\leq 1}$ for all ${\displaystyle i}$, and then ${\displaystyle \textstyle \sum |x_{i}|^{q}\leq \textstyle \sum |x_{i}|^{p}=1}$.

ℓ 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 or ${\displaystyle \,\aleph _{0}\,}$). The following two items are related:

• If H is infinite dimensional, then it is isomorphic to ℓ
• If dim(H) = N, then H is isomorphic to ${\displaystyle \mathbb {C} ^{N}}$

Properties of ℓ spaces

A sequence of elements in ℓ converges in the space of complex sequences ℓ if and only if it converges weakly in this space. If K is a subset of this space, then the following are equivalent:

1. K is compact;
2. K is weakly compact;
3. K is bounded, closed, and equismall at infinity.

Here K being equismall at infinity means that for every ${\displaystyle \varepsilon >0}$, there exists a natural number ${\displaystyle n_{\varepsilon }\geq 0}$ such that ${\textstyle \sum _{n=n_{\epsilon }}^{\infty }|s_{n}|<\varepsilon }$ for all ${\displaystyle s=\left(s_{n}\right)_{n=1}^{\infty }\in K}$.

See also

• L space
• Tsirelson space
• beta-dual space
• Orlicz sequence space
• Hilbert space

References

1. ^ Jarchow 1981, pp. 129–130.
2. Debnath, Lokenath; Mikusinski, Piotr (2005). Hilbert Spaces with Applications. Elsevier. pp. 120–121. ISBN 978-0-12-2084386.
3. ^ Trèves 2006, pp. 451–458.

Bibliography

• Banach, Stefan; Mazur, S. (1933), "Zur Theorie der linearen Dimension", Studia Mathematica, 4: 100–112.
• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Pitt, H.R. (1936), "A note on bilinear forms", J. London Math. Soc., 11 (3): 174–180, doi:10.1112/jlms/s1-11.3.174.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schur, J. (1921), "Über lineare Transformationen in der Theorie der unendlichen Reihen", Journal für die reine und angewandte Mathematik, 151: 79–111, doi:10.1515/crll.1921.151.79.
• Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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L spaces	Integrable function Lebesgue integration Taxicab geometry
L spaces	Bessel's Cauchy–Schwarz Euclidean distance Hilbert space Parseval's identity Polarization identity Pythagorean theorem Square-integrable function
${\displaystyle L^{\infty }}$ spaces	Bounded function Chebyshev distance Infimum and supremum Essential Uniform norm
Maps	Almost everywhere Convergence almost everywhere Convergence in measure Function space Integral transform Locally integrable function Measurable function Symmetric decreasing rearrangement
Inequalities	Babenko–Beckner Chebyshev's Clarkson's Hanner's Hausdorff–Young Hölder's Markov's Minkowski Young's convolution
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Applications & related	Bochner space Fourier analysis Lorentz space Probability theory Quasinorm Real analysis Sobolev space *-algebra C*-algebra Von Neumann

• Sequence spaces
• Functional analysis
• Sequences and series