Sequence space

For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic.

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 $x_{\bullet }=\left(x_{n}\right)_{n\in \mathbb {N} }$ in a set $X$ is just an $X$-valued map $x_{\bullet }:\mathbb {N} \to X$ whose value at $n\in \mathbb {N}$ is denoted by $x_{n}$ instead of the usual parentheses notation $x(n).$

Space of all sequences

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

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

and componentwise scalar multiplication

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

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

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

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

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

But the product topology is also unavoidable: $\mathbb {K} ^{\mathbb {N} }$ 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^p spaces

See also: L^p space and L-infinity

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

$$\sum _{n}|x_{n}|^{p}<\infty .$$

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

$$\|x\|_{p}~=~\left(\sum _{n}|x_{n}|^{p}\right)^{1/p}\qquad {\text{ for all }}x\in \ell ^{p}$$

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.

If $p=2$ then $\ell ^{2}$ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all $x_{\bullet },y_{\bullet }\in \ell ^{p}$ by

$$\langle x_{\bullet },y_{\bullet }\rangle ~=~\sum _{n}{\overline {x_{n}}}y_{n}.$$

The canonical norm induced by this inner product is the usual $\ell ^{2}$-norm, meaning that $\|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}$ for all $\mathbf {x} \in \ell ^{p}.$

If $p=\infty ,$ then $\ell ^{\infty }$ is defined to be the space of all bounded sequences endowed with the norm

$$\|x\|_{\infty }~=~\sup _{n}|x_{n}|,$$

$\ell ^{\infty }$ is also a Banach space.

If $0<p<1,$ then $\ell ^{p}$ does not carry a norm, but rather a metric defined by

$$d(x,y)~=~\sum _{n}\left|x_{n}-y_{n}\right|^{p}.\,$$

c, c₀ and c₀₀

See also: c space

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

A sequence that converges to $0$ is called a null sequence and is said to vanish. 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 $c_{0}$ and is called the space of null sequences or the space of vanishing sequences.

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

Space of all finite sequences

Let

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

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

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

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

The image of each inclusion is

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

and consequently,

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

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

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

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

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

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

Other sequence spaces

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

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

This space, when equipped with the norm

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

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

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

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 $c_{00}$ 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

See also: c space

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

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

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

$$L_{x}(y)=\sum _{n}x_{n}y_{n}$$

for y in ℓ^p. Hölder's inequality implies that L_x is a bounded linear functional on ℓ^p, and in fact

$$|L_{x}(y)|\leq \|x\|_{q}\,\|y\|_{p}$$

so that the operator norm satisfies

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

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

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

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

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

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 $x\mapsto L_{x}$ gives an isometry

$$\kappa _{q}:\ell ^{q}\to (\ell ^{p})^{*}.$$

The map

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

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 $Q:\ell ^{1}\to X$, 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.

ℓ^p spaces are increasing in p

For $p\in [1,\infty ]$, the spaces $\ell ^{p}$ are increasing in $p$, with the inclusion operator being continuous: for $1\leq p<q\leq \infty$, one has $\|x\|_{q}\leq \|x\|_{p}$. 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 $\textstyle \sum |x_{i}|^{q}\leq 1$ for $q>p$. But if $\|x\|_{p}=1$, then $|x_{i}|\leq 1$ for all $i$, and then $\textstyle \sum |x_{i}|^{q}\leq \textstyle \sum |x_{i}|^{p}=1$.

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 $\,\aleph _{0}\,$). 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 $\mathbb {C} ^{N}$

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

See also

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

Bibliography

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