Sequence space

A sequence \textstyle x_{\bull} = (x_n)_{n \in \N} in a set is an -valued map x_{\bull} : \N \to X whose value at is denoted by instead of the usual parentheses notation . Space of all sequences Let denote the field either of real or complex numbers. The set {{tmath|\textstyle \mathbb{K}^\N}} of all sequences of elements of 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 {{tmath|\textstyle \mathbb{K}^\N}}. As a topological space, {{tmath|\textstyle \mathbb{K}^\N}} is naturally endowed with the product topology. Under this topology, {{tmath|\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 {{tmath|\textstyle \mathbb{K}^\N}} (and thus the product topology cannot be defined by any norm). Among Fréchet spaces, {{tmath|\textstyle \mathbb{K}^\N}} is minimal in having no continuous norms: {{Math theorem Then the following are equivalent: admits no continuous norm (that is, any continuous seminorm on has a nontrivial null space). contains a vector subspace TVS-isomorphic to {{tmath|\textstyle \mathbb{K}^\N}}. contains a complemented vector subspace TVS-isomorphic to {{tmath|\textstyle \mathbb{K}^\N}}. }}But the product topology is also unavoidable: {{tmath|\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. spaces For , is the subspace of {{tmath|\textstyle \mathbb{K}^\N}} consisting of all sequences \textstyle x_{\bull} = (x_n)_{n \in \N} satisfying \sum_n |x_n|^p If , then the real-valued function \|\cdot\|_p on 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 . In fact, is a complete metric space with respect to this norm, and therefore is a Banach space. If then is also a Hilbert space when endowed with its canonical inner product, called the '''''', defined for all 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 -norm, meaning that \textstyle \|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle} for all {{tmath|\textstyle \mathbf{x} \in \ell^p}}. If , then is defined to be the space of all bounded sequences endowed with the norm \|x\|_\infty ~=~ \sup_n |x_n|, is also a Banach space. If , then does not carry a norm, but rather a metric defined by d(x,y) ~=~ \sum_n \left|x_n - y_n\right|^p. , and A is any sequence \textstyle x_{\bull} \in \mathbb{K}^\N such that \textstyle \lim_{n \to \infty} x_n exists. The set of all convergent sequences is a vector subspace of {{tmath|\textstyle \mathbb{K}^\N\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}. Space of all finite sequences 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 . As a vector space, \textstyle \mathbb{K}^\infty is equal to {{tmath|c_{00} }}, but {{tmath|\textstyle \mathbb{K}^\infty}} has a different topology. For every natural number , let {{tmath|\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 {{tmath|\textstyle \mathbb{K}^\infty}} a final topology , defined to be the finest topology on {{tmath|\textstyle \mathbb{K}^\infty}} such that all the inclusions are continuous (an example of a coherent topology). With this topology, {{tmath|\textstyle \mathbb{K}^\infty}} becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is Fréchet–Urysohn. The topology is also strictly finer than the subspace topology induced on {{tmath|\textstyle \mathbb{K}^\infty}} by {{tmath|\textstyle \mathbb{K}^\N}}. Convergence in has a natural description: if \textstyle v \in \mathbb{K}^\infty and is a sequence in {{tmath|\textstyle \mathbb{K}^\infty}} then in if and only is eventually contained in a single image \textstyle \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right) and under the natural topology of that image. Often, each image \textstyle \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right) is identified with the corresponding {{tmath|\textstyle \mathbb{K}^n}}; explicitly, the elements \textstyle \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 {{nobr|\textstyle \operatorname{Im} \left( \operatorname{In}_{\mathbb{K}^n} \right),}} the quotient topology from the map {{nobr|\textstyle \operatorname{In}_{\mathbb{K}^n},}} and the Euclidean topology on {{tmath|\textstyle \mathbb{K}^n}} all coincide. With this identification, \textstyle \left( \left(\mathbb{K}^\infty, \tau^\infty\right), \left(\operatorname{In}_{\mathbb{K}^n}\right)_{n \in \N}\right) is the direct limit of the directed system \textstyle \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), 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 \textstyle \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 for which \sup_n \biggl\vert \sum_{i=0}^n x_i \biggr\vert 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 under this isomorphism. The space 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 spaces and the space ==

The space is the only 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 and directly shows that the identity is not true unless . Each is distinct, in that is a strict subset of whenever ; furthermore, is not linearly isomorphic to when . In fact, by Pitt's theorem , every bounded linear operator from to is compact when . No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of , and is thus said to be strictly singular. If , then the (continuous) dual space of is isometrically isomorphic to , where is the Hölder conjugate of : . The specific isomorphism associates to an element of the functional L_x(y) = \sum_n x_n y_n for in . Hölder's inequality implies that is a bounded linear functional on , 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{\|y\|_p} \le \|x\|_q. In fact, taking to be the element of 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 so that in fact \|L_x\|_{(\ell^p)^*} = \|x\|_q. Conversely, given a bounded linear functional on , the sequence defined by lies in . Thus the mapping 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 with the inverse of its transpose coincides with the canonical injection of into its double dual. As a consequence is a reflexive space. By abuse of notation, it is typical to identify with the dual of : . Then reflexivity is understood by the sequence of identifications {{tmath|1=\textstyle (\ell^p)^{**} = (\ell^q)^* = \ell^p}}. The space is defined as the space of all sequences converging to zero, with norm identical to It is a closed subspace of , hence a Banach space. The dual of is ; the dual of is . For the case of natural numbers index set, the and are separable, with the sole exception of . The dual of is the ba space. The spaces and (for ) have a canonical unconditional Schauder basis {{tmath|1= \{e_i : i = 1, 2, \ldots \} }}, where is the sequence which is zero but for a in the th entry. The space ℓ1 has the Schur property: In ℓ1, any sequence that is weakly convergent is also strongly convergent . 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 spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some or of , 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 , was answered in the affirmative by . That is, for every separable Banach space , there exists a quotient map , so that is isomorphic to . In general, {{tmath| \operatorname{ker} Q}} is not complemented in , that is, there does not exist a subspace of such that . In fact, has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ; since there are uncountably many such s, and since no is isomorphic to any other, there are thus uncountably many ker Qs). Except for the trivial finite-dimensional case, an unusual feature of is that it is not polynomially reflexive. spaces are increasing in For , the spaces are increasing in , with the inclusion operator being continuous: for , one has Indeed, the inequality is homogeneous in the , so it is sufficient to prove it under the assumption that In this case, we need only show that \textstyle\sum |x_i|^q \le 1 for . But if then |x_i|\le 1 for all , and then \textstyle \sum |x_i|^q \le {}\! is isomorphic to all separable, infinite dimensional Hilbert spaces Let be a separable Hilbert space. Every orthogonal set in is at most countable (i.e. has finite dimension or ). The following two items are related: • If is infinite dimensional, then it is isomorphic to , • If {{tmath|1= \operatorname{dim}(H) = N}}, then is isomorphic to . == Properties of spaces ==

A sequence of elements in converges in the space of complex sequences if and only if it converges weakly in this space. If is a subset of this space, then the following are equivalent: • is compact; • is weakly compact; • is bounded, closed, and equismall at infinity. Here being equismall at infinity means that for every , there exists a natural number n_{\varepsilon} \geq 0 such that \textstyle \sum_{n = n_{\epsilon}}^\infty | s_n | for all {{tmath|1=\textstyle s = \left( s_n \right)_{n=1}^\infty \in K}}. == See also ==