Sequence space

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^p spaces, consisting of the p-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 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).^[1] Among Fréchet spaces, ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ is minimal in having no continuous norms:

But the product topology is also unavoidable: ${\displaystyle \mathbb {K} ^{\mathbb {N} }}$ does not admit a strictly coarser Hausdorff, locally convex topology.^[1] 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^p 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

If ${\displaystyle p\geq 1,}$ then the real-valued function ${\displaystyle \|\cdot \|_{p}}$ on ${\displaystyle \ell ^{p}}$ defined by

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

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

${\displaystyle \ell ^{\infty }}$ 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

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^p spaces and the space c₀

The space ℓ² is the only ℓ^p 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^p is distinct, in that ℓ^p is a strict subset of ℓ^s whenever p < s; furthermore, ℓ^p is not linearly isomorphic to ℓ^s when p ≠ s. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from ℓ^s to ℓ^p 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^s, 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 ℓ^q the functional

for y in ℓ^p. Hölder's inequality implies that L_x is a bounded linear functional on ℓ^p, and in fact 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^p 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^p, the sequence defined by x_n = L(e_n) lies in ℓ^q. Thus the mapping ${\displaystyle x\mapsto L_{x}}$ gives an isometry

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^q into its double dual. As a consequence ℓ^q is a reflexive space. 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 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 ℓ¹; the dual of ℓ¹ is ℓ^∞. For the case of natural numbers index set, the ℓ^p and c₀ are separable, with the sole exception of ℓ^∞. The dual of ℓ^∞ is the ba space.

The spaces c₀ and ℓ^p (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^th entry.

The space ℓ¹ has the Schur property: In ℓ¹, 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 ℓ¹ that are weak convergent but not strong convergent.

The ℓ^p spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ^p 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 ℓ¹, 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 ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that ${\displaystyle \ell ^{1}=Y\oplus \ker Q}$. In fact, ℓ¹ 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^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.

ℓ^p spaces are increasing in p

For ${\displaystyle p\in [1,\infty ]}$, 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 ${\displaystyle \textstyle \sum |x_{i}|^{q}\leq \textstyle \sum |x_{i}|^{p}=1}$.

ℓ² 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 ${\displaystyle \,\aleph _{0}\,}$).^[2] The following two items are related:

• If H is infinite dimensional, then it is isomorphic to ℓ²
• If dim(H) = N, then H is isomorphic to ${\displaystyle \mathbb {C} ^{N}}$

Properties of ℓ¹ spaces

A sequence of elements in ℓ¹ converges in the space of complex sequences ℓ¹ if and only if it converges weakly in this space.^[3] If K is a subset of this space, then the following are equivalent:^[3]

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^p space
• Tsirelson space
• beta-dual space
• Orlicz sequence space
• Hilbert space

References

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.
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• 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.
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• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.