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 ⁠\(\mathbb K\)⁠ 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 ⁠\(\mathbb K\)⁠, 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 ⁠\(\textstyle \ell^p\)⁠ 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 ⁠\(c_0\)⁠, 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.

A sequence \(\textstyle x_{\bull} = (x_n)_{n \in \N}\) in a set ⁠\(X\)⁠ is an ⁠\(X\)⁠-valued map \(x_{\bull} : \N \to X\) whose value at ⁠\(n \in \N\)⁠ is denoted by ⁠\(x_n\)⁠ instead of the usual parentheses notation ⁠\(x(n)\)⁠.

Let ⁠\(\mathbb K\)⁠ denote the field either of real or complex numbers. The set ⁠\(\textstyle \mathbb{K}^\N\)⁠ of all sequences of elements of ⁠\(\mathbb K\)⁠ is a vector space for componentwise addition \[\left(x_n\right)_{n \in \N} + \left(y_n\right)_{n \in \N} = \left(x_n + y_n\right)_{n \in \N},\] and componentwise scalar multiplication \[\alpha\left(x_n\right)_{n \in \N} = \left(\alpha x_n\right)_{n \in \N}.\]

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

As a topological space, ⁠\(\textstyle \mathbb{K}^\N\)⁠ is naturally endowed with the product topology. Under this topology, ⁠\(\textstyle \mathbb{K}^\N\)⁠ 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 ⁠\(\textstyle \mathbb{K}^\N\)⁠ (and thus the product topology cannot be defined by any norm). Among Fréchet spaces, ⁠\(\textstyle \mathbb{K}^\N\)⁠ 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 ⁠\(\textstyle \mathbb{K}^\N\)⁠.
3. ⁠\(X\)⁠ contains a complemented vector subspace TVS-isomorphic to ⁠\(\textstyle \mathbb{K}^\N\)⁠.

But the product topology is also unavoidable: ⁠\(\textstyle \mathbb{K}^\N\)⁠ 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.

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

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

If ⁠\(p = 2\)⁠ then ⁠\(\textstyle \ell^2\)⁠ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all ⁠\(\textstyle x_\bull, y_\bull \in \ell^p\)⁠ by \[\langle x_\bull, y_\bull \rangle ~=~ \sum_n \overline{x_n\!}\, y_n.\] The canonical norm induced by this inner product is the usual ⁠\(\textstyle \ell^2\)⁠-norm, meaning that \(\textstyle \|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}\) for all ⁠\(\textstyle \mathbf{x} \in \ell^p\)⁠.

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

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

A convergent sequence is any sequence \(\textstyle x_{\bull} \in \mathbb{K}^\N\) such that \(\textstyle \lim_{n \to \infty} x_n\) exists. The set ⁠\(c\)⁠ of all convergent sequences is a vector subspace of ⁠\(\textstyle \mathbb{K}^\N\)⁠ 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 ⁠\(\textstyle \ell^\infty\)⁠ 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 \(\textstyle (x_{nk})_{k \in \N}\) where \(x_{nk} = 1/k\) for the first \(n\) entries (for \(k = 1, \ldots, n\)) and is zero everywhere else (that is, \(\textstyle (x_{nk})_{k \in \N} = {}\!\)\(\bigl(1, \tfrac12, \ldots,{}\)\(\tfrac{1}{n-1}, \tfrac{1}{n}, {}\)\(0, 0, \ldots\bigr)\)) is a Cauchy sequence but it does not converge to a sequence in \(c_{00}.\)

Let \[\mathbb{K}^\infty=\left\{\left(x_1, x_2,\ldots\right)\in\mathbb{K}^\N : \text{all but finitely many }x_i\text{ equal }0\right\}\]

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

For every natural number ⁠\(n \in \N\)⁠, let ⁠\(\textstyle \mathbb{K}^n\)⁠ denote the usual Euclidean space endowed with the Euclidean topology and let \(\textstyle \operatorname{In}_{\mathbb{K}^n} : \mathbb{K}^n \to \mathbb{K}^\infty\) 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, \[\mathbb{K}^\infty = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right).\]

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

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

Condensed: the full section is in Wikipedia.

The space of bounded series, denote by bs, is the space of sequences ⁠\(x\)⁠ for which \[\sup_n \biggl\vert \sum_{i=0}^n x_i \biggr\vert < \infty.\]

This space, when equipped with the norm \[\|x\|_{bs} = \sup_n \biggl\vert \sum_{i=0}^n x_i \biggr\vert,\]

is a Banach space isometrically isomorphic to \(\textstyle \ell^\infty,\) via the linear mapping \[(x_n)_{n \in \N} \mapsto \biggl(\sum_{i=0}^n x_i\biggr)_{n \in \N}.\]

The subspace \(cs\) consisting of all convergent series is a subspace that goes over to the space ⁠\(c\)⁠ under this isomorphism.

The space ⁠\(\Phi\)⁠ 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.

The space ⁠\(\textstyle \ell^2\)⁠ is the only ⁠\(\textstyle \ell^p\)⁠ 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 ⁠\(\textstyle \ell^p\)⁠ is distinct, in that ⁠\(\textstyle \ell^p\)⁠ is a strict subset of ⁠\(\textstyle \ell^s\)⁠ whenever ⁠\(p < s\)⁠; furthermore, ⁠\(\textstyle \ell^p\)⁠ is not linearly isomorphic to ⁠\(\textstyle \ell^s\)⁠ when ⁠\(p \neq s\)⁠. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from ⁠\(\textstyle \ell^s\)⁠ to ⁠\(\textstyle \ell^p\)⁠ 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 ⁠\(\ell^s\)⁠, and is thus said to be strictly singular.

If ⁠\(1 < p < \infty\)⁠, then the (continuous) dual space of ⁠\(\textstyle \ell^p\)⁠ is isometrically isomorphic to ⁠\(\textstyle \ell^q\)⁠, where ⁠\(q\)⁠ is the Hölder conjugate of ⁠\(p\)⁠: ⁠\(1/p + 1/q = 1\)⁠. The specific isomorphism associates to an element ⁠\(x\)⁠ of ⁠\(\textstyle \ell^q\)⁠ the functional \[L_x(y) = \sum_n x_n y_n\] for ⁠\(y\)⁠ in ⁠\(\textstyle \ell^p\)⁠. Hölder's inequality implies that ⁠\(L_x\)⁠ is a bounded linear functional on ⁠\(\textstyle \ell^p\)⁠, and in fact \[|L_x(y)| \le \|x\|_q\, \|y\|_p\] so that the operator norm satisfies \[\|L_x\|_{(\ell^p)^*} \mathrel{\stackrel{\rm{def}}{=}} \sup_{y\in\ell^p, y\not=0} \frac{|L_x(y)|}{\|y\|_p} \le \|x\|_q.\] In fact, taking ⁠\(y\)⁠ to be the element of ⁠\(\textstyle \ell^p\)⁠ 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 ⁠\(\textstyle \ell^p\)⁠, the sequence defined by ⁠\(x_n = L(e_n)\)⁠ lies in ⁠\(\textstyle \ell^q\)⁠. 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 ⁠\(\kappa_p\)⁠ with the inverse of its transpose coincides with the canonical injection of ⁠\(\textstyle \ell^q\)⁠ into its double dual. As a consequence ⁠\(\textstyle \ell^q\)⁠ is a reflexive space. By abuse of notation, it is typical to identify ⁠\(\textstyle \ell^q\)⁠ with the dual of ⁠\(\textstyle \ell^p\)⁠: ⁠\(\textstyle (\ell^p)^* = \ell^q\)⁠. Then reflexivity is understood by the sequence of identifications ⁠\(\textstyle (\ell^p)^{**} = (\ell^q)^* = \ell^p\)⁠.

The space ⁠\(c_0\)⁠ is defined as the space of all sequences converging to zero, with norm identical to \(\|x\|_\infty\). It is a closed subspace of ⁠\(\textstyle \ell^\infty\)⁠, hence a Banach space. The dual of ⁠\(c_0\)⁠ is ⁠\(\textstyle \ell^1\)⁠; the dual of ⁠\(\textstyle \ell^1\)⁠ is ⁠\(\textstyle \ell^\infty\)⁠. For the case of natural numbers index set, the ⁠\(\textstyle \ell^p\)⁠ and ⁠\(c_0\)⁠ are separable, with the sole exception of ⁠\(\textstyle \ell^\infty\)⁠. The dual of ⁠\(\textstyle \ell^\infty\)⁠ is the ba space.

Condensed: the full section is in Wikipedia.

For ⁠\(p \in [1,\infty]\)⁠, the spaces ⁠\(\textstyle \ell^p\)⁠ are increasing in ⁠\(p\)⁠, with the inclusion operator being continuous: for ⁠\(1 \le p < q \le \infty\)⁠, one has \(\|x\|_q\le\|x\|_p\). 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 \(\textstyle\sum |x_i|^q \le 1\) for ⁠\(q > p\)⁠. But if \(\|x\|_p = 1\), then \(|x_i|\le 1\) for all ⁠\(i\)⁠, and then \(\textstyle \sum |x_i|^q \le {}\!\)\(\textstyle\sum |x_i|^p = 1\).

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 ⁠\(\textstyle \ell^2\)⁠,
• If ⁠\(\operatorname{dim}(H) = N\)⁠, then ⁠\(H\)⁠ is isomorphic to ⁠\(\textstyle \C^N\)⁠.

A sequence of elements in ⁠\(\textstyle \ell^1\)⁠ converges in the space of complex sequences ⁠\(\textstyle \ell^1\)⁠ 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 ⁠\(\textstyle s = \left( s_n \right)_{n=1}^\infty \in K\)⁠.