Sequence space

Short description: Vector space of infinite sequences

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

Space of all sequences

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

[math]\displaystyle{ \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]\displaystyle{ \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 [math]\displaystyle{ \mathbb{K}^{\N}. }[/math]

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

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

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

But the product topology is also unavoidable: [math]\displaystyle{ \mathbb{K}^{\N} }[/math] 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

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

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

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

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

If [math]\displaystyle{ 0 \lt p \lt 1, }[/math] then [math]\displaystyle{ \ell^p }[/math] does not carry a norm, but rather a metric defined by [math]\displaystyle{ d(x,y) ~=~ \sum_n \left|x_n - y_n\right|^p.\, }[/math]

c, c₀ and c₀₀

A convergent sequence is any sequence [math]\displaystyle{ x_{\bull} \in \mathbb{K}^{\N} }[/math] such that [math]\displaystyle{ \lim_{n \to \infty} x_n }[/math] exists. The set [math]\displaystyle{ c }[/math] of all convergent sequences is a vector subspace of [math]\displaystyle{ \mathbb{K}^{\N} }[/math] called the space of convergent sequences. Since every convergent sequence is bounded, [math]\displaystyle{ c }[/math] is a linear subspace of [math]\displaystyle{ \ell^{\infty}. }[/math] Moreover, this sequence space is a closed subspace of [math]\displaystyle{ \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]\displaystyle{ 0 }[/math] is called a null sequence and is said to vanish. The set of all sequences that converge to [math]\displaystyle{ 0 }[/math] is a closed vector subspace of [math]\displaystyle{ c }[/math] that when endowed with the supremum norm becomes a Banach space that is denoted by [math]\displaystyle{ c_0 }[/math] and is called the space of null sequences or the space of vanishing sequences.

The space of eventually zero sequences, [math]\displaystyle{ c_{00}, }[/math] is the subspace of [math]\displaystyle{ 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]\displaystyle{ \left(x_{nk}\right)_{k \in \N} }[/math] where [math]\displaystyle{ x_{nk} = 1/k }[/math] for the first [math]\displaystyle{ n }[/math] entries (for [math]\displaystyle{ k = 1, \ldots, n }[/math]) and is zero everywhere else (that is, [math]\displaystyle{ \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]\displaystyle{ c_{00}. }[/math]

Space of all finite sequences

Let

[math]\displaystyle{ \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]\displaystyle{ \mathbb{K} }[/math]. As a vector space, [math]\displaystyle{ \mathbb{K}^{\infty} }[/math] is equal to [math]\displaystyle{ c_{00} }[/math], but [math]\displaystyle{ \mathbb{K}^{\infty} }[/math] has a different topology.

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

[math]\displaystyle{ \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]\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\} }[/math]

and consequently,

[math]\displaystyle{ \mathbb{K}^{\infty} = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right). }[/math]

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

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

Often, each image [math]\displaystyle{ \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right) }[/math] is identified with the corresponding [math]\displaystyle{ \mathbb{K}^n }[/math]; explicitly, the elements [math]\displaystyle{ \left( x_1, \ldots, x_n \right) \in \mathbb{K}^n }[/math] and [math]\displaystyle{ \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]\displaystyle{ \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right) }[/math], the quotient topology from the map [math]\displaystyle{ \operatorname{In}_{\mathbb{K}^n} }[/math], and the Euclidean topology on [math]\displaystyle{ \mathbb{K}^n }[/math] all coincide. With this identification, [math]\displaystyle{ \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]\displaystyle{ \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]\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) }[/math].

This shows [math]\displaystyle{ \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 [math]\displaystyle{ x }[/math] for which

[math]\displaystyle{ \sup_n \left\vert \sum_{i=0}^n x_i \right\vert \lt \infty. }[/math]

This space, when equipped with the norm

[math]\displaystyle{ \|x\|_{bs} = \sup_n \left\vert \sum_{i=0}^n x_i \right\vert, }[/math]

is a Banach space isometrically isomorphic to [math]\displaystyle{ \ell^{\infty}, }[/math] via the linear mapping

[math]\displaystyle{ (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]\displaystyle{ 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 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

[math]\displaystyle{ \|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 = 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 [math]\displaystyle{ L_x(y) = \sum_n x_n y_n }[/math] for y in ℓ^p. Hölder's inequality implies that L_x is a bounded linear functional on ℓ^p, and in fact [math]\displaystyle{ |L_x(y)| \le \|x\|_q\,\|y\|_p }[/math] so that the operator norm satisfies

[math]\displaystyle{ \|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 ℓ^p with

[math]\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} }[/math]

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

[math]\displaystyle{ \|L_x\|_{(\ell^p)^*} = \|x\|_q. }[/math]

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 [math]\displaystyle{ x\mapsto L_x }[/math] gives an isometry [math]\displaystyle{ \kappa_q : \ell^q \to (\ell^p)^*. }[/math]

The map

[math]\displaystyle{ \ell^q\xrightarrow{\kappa_q}(\ell^p)^*\xrightarrow{(\kappa_q^*)^{-1}} }[/math]

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). That is, for every separable Banach space X, there exists a quotient map [math]\displaystyle{ Q:\ell^1 \to X }[/math], so that X is isomorphic to [math]\displaystyle{ \ell^1 / \ker Q }[/math]. In general, ker Q is not complemented in ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that [math]\displaystyle{ \ell^1 = Y \oplus \ker Q }[/math]. In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take [math]\displaystyle{ X=\ell^p }[/math]; 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 [math]\displaystyle{ p\in[1,\infty] }[/math], the spaces [math]\displaystyle{ \ell^p }[/math] are increasing in [math]\displaystyle{ p }[/math], with the inclusion operator being continuous: for [math]\displaystyle{ 1\le p\lt q\le\infty }[/math], one has [math]\displaystyle{ \|x\|_q\le\|x\|_p }[/math]. Indeed, the inequality is homogeneous in the [math]\displaystyle{ x_i }[/math], so it is sufficient to prove it under the assumption that [math]\displaystyle{ \|x\|_p = 1 }[/math]. In this case, we need only show that [math]\displaystyle{ \textstyle\sum |x_i|^q \le 1 }[/math] for [math]\displaystyle{ q\gt p }[/math]. But if [math]\displaystyle{ \|x\|_p = 1 }[/math], then [math]\displaystyle{ |x_i|\le 1 }[/math] for all [math]\displaystyle{ i }[/math], and then [math]\displaystyle{ \textstyle\sum |x_i|^q \le \textstyle\sum |x_i|^p = 1 }[/math].

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 [math]\displaystyle{ \,\aleph_0\, }[/math]). 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 [math]\displaystyle{ \Complex^N }[/math]

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 [math]\displaystyle{ \varepsilon \gt 0 }[/math], there exists a natural number [math]\displaystyle{ n_{\varepsilon} \geq 0 }[/math] such that [math]\displaystyle{ \sum_{n = n_{\epsilon}}^{\infty} | s_n | \lt \varepsilon }[/math] for all [math]\displaystyle{ s = \left( s_n \right)_{n=1}^{\infty} \in K }[/math].

See also

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

Bibliography

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