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 ⁠${\displaystyle \mathbb {K} }$ ⁠ 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 ⁠${\displaystyle \mathbb {K} }$ ⁠, 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 ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ spaces, consisting of the ⁠${\displaystyle p}$ ⁠-power summable sequences, with the ⁠${\displaystyle p}$ ⁠-norm. These are special cases of ⁠${\displaystyle L^{p}}$ ⁠ 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 ⁠${\displaystyle c}$ ⁠ and ⁠${\displaystyle c_{0}}$ ⁠, 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.

Definition

A sequence ${\displaystyle \textstyle x_{\bullet }=(x_{n})_{n\in \mathbb {N} }}$ in a set ⁠${\displaystyle X}$ ⁠ is 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 ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ 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 ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠.

As a topological space, ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ is naturally endowed with the product topology. Under this topology, ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ 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 ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ (and thus the product topology cannot be defined by any norm).^[1] Among Fréchet spaces, ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ is minimal in having no continuous norms:

Theorem^[1]—Let ⁠${\displaystyle X}$ ⁠ be a Fréchet space over ⁠${\displaystyle \mathbb {K} }$ ⁠. Then the following are equivalent:

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

But the product topology is also unavoidable: ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ 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 ⁠${\displaystyle 0<p<\infty }$ ⁠, ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is the subspace of ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠ consisting of all sequences ${\displaystyle \textstyle x_{\bullet }=(x_{n})_{n\in \mathbb {N} }}$ satisfying $${\displaystyle \sum _{n}|x_{n}|^{p}<\infty .}$$

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

If ⁠${\displaystyle p=2}$ ⁠ then ⁠${\displaystyle \textstyle \ell ^{2}}$ ⁠ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all ⁠${\displaystyle \textstyle x_{\bullet },y_{\bullet }\in \ell ^{p}}$ ⁠ by $${\displaystyle \langle x_{\bullet },y_{\bullet }\rangle ~=~\sum _{n}{\overline {x_{n}\!}}\,y_{n}.}$$ The canonical norm induced by this inner product is the usual ⁠${\displaystyle \textstyle \ell ^{2}}$ ⁠-norm, meaning that ${\displaystyle \textstyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}$ for all ⁠${\displaystyle \textstyle \mathbf {x} \in \ell ^{p}}$ ⁠.

If ⁠${\displaystyle p=\infty }$ ⁠, then ⁠${\displaystyle \textstyle \ell ^{\infty }}$ ⁠ is defined to be the space of all bounded sequences endowed with the norm $${\displaystyle \|x\|_{\infty }~=~\sup _{n}|x_{n}|,}$$ ⁠${\displaystyle \textstyle \ell ^{\infty }}$ ⁠ is also a Banach space.

If ⁠${\displaystyle 0<p<1}$ ⁠, then ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ does not carry a norm, but rather a metric defined by $${\displaystyle d(x,y)~=~\sum _{n}\left|x_{n}-y_{n}\right|^{p}.}$$

c, c₀ and c₀₀

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

A sequence that converges to ⁠${\displaystyle 0}$ ⁠ 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 ${\displaystyle \textstyle (x_{nk})_{k\in \mathbb {N} }}$ 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, ${\displaystyle \textstyle (x_{nk})_{k\in \mathbb {N} }={}\!}$ ${\displaystyle {\bigl (}1,{\tfrac {1}{2}},\ldots ,{}}$ ${\displaystyle {\tfrac {1}{n-1}},{\tfrac {1}{n}},{}}$ ${\displaystyle 0,0,\ldots {\bigr )}}$ ) 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, ${\displaystyle \textstyle \mathbb {K} ^{\infty }}$ is equal to ⁠${\displaystyle c_{00}}$ ⁠, but ⁠${\displaystyle \textstyle \mathbb {K} ^{\infty }}$ ⁠ has a different topology.

For every natural number ⁠${\displaystyle n\in \mathbb {N} }$ ⁠, let ⁠${\displaystyle \textstyle \mathbb {K} ^{n}}$ ⁠ denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \textstyle \operatorname {In} _{\mathbb {K} ^{n}}:\mathbb {K} ^{n}\to \mathbb {K} ^{\infty }}$ 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 ⁠${\displaystyle \textstyle \mathbb {K} ^{\infty }}$ ⁠ a final topology ⁠${\displaystyle \textstyle \tau ^{\infty }}$ ⁠, defined to be the finest topology on ⁠${\displaystyle \textstyle \mathbb {K} ^{\infty }}$ ⁠ such that all the inclusions are continuous (an example of a coherent topology). With this topology, ⁠${\displaystyle \textstyle \mathbb {K} ^{\infty }}$ ⁠ becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is not Fréchet–Urysohn. The topology ⁠${\displaystyle \textstyle \tau ^{\infty }}$ ⁠ is also strictly finer than the subspace topology induced on ⁠${\displaystyle \textstyle \mathbb {K} ^{\infty }}$ ⁠ by ⁠${\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}$ ⁠.

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

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

Other sequence spaces

The space of bounded series, denote by bs, is the space of sequences ⁠${\displaystyle x}$ ⁠ for which $${\displaystyle \sup _{n}{\biggl \vert }\sum _{i=0}^{n}x_{i}{\biggr \vert }<\infty .}$$

This space, when equipped with the norm $${\displaystyle \|x\|_{bs}=\sup _{n}{\biggl \vert }\sum _{i=0}^{n}x_{i}{\biggr \vert },}$$

is a Banach space isometrically isomorphic to ${\displaystyle \textstyle \ell ^{\infty },}$ via the linear mapping $${\displaystyle (x_{n})_{n\in \mathbb {N} }\mapsto {\biggl (}\sum _{i=0}^{n}x_{i}{\biggr )}_{n\in \mathbb {N} }.}$$

The subspace ${\displaystyle cs}$ consisting of all convergent series is a subspace that goes over to the space ⁠${\displaystyle c}$ ⁠ under this isomorphism.

The space ⁠${\displaystyle \Phi }$ ⁠ 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 ⁠${\displaystyle \textstyle \ell ^{2}}$ ⁠ is the only ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ 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 ⁠${\displaystyle x}$ ⁠ and ⁠${\displaystyle y}$ ⁠ directly shows that the identity is not true unless ⁠${\displaystyle p=2}$ ⁠.

Each ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is distinct, in that ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is a strict subset of ⁠${\displaystyle \textstyle \ell ^{s}}$ ⁠ whenever ⁠${\displaystyle p<s}$ ⁠; furthermore, ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is not linearly isomorphic to ⁠${\displaystyle \textstyle \ell ^{s}}$ ⁠ when ⁠${\displaystyle p\neq s}$ ⁠. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from ⁠${\displaystyle \textstyle \ell ^{s}}$ ⁠ to ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is compact when ⁠${\displaystyle p<s}$ ⁠. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ⁠${\displaystyle \ell ^{s}}$ ⁠, and is thus said to be strictly singular.

If ⁠${\displaystyle 1<p<\infty }$ ⁠, then the (continuous) dual space of ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is isometrically isomorphic to ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠, where ⁠${\displaystyle q}$ ⁠ is the Hölder conjugate of ⁠${\displaystyle p}$ ⁠: ⁠${\displaystyle 1/p+1/q=1}$ ⁠. The specific isomorphism associates to an element ⁠${\displaystyle x}$ ⁠ of ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠ the functional $${\displaystyle L_{x}(y)=\sum _{n}x_{n}y_{n}}$$ for ⁠${\displaystyle y}$ ⁠ in ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠. Hölder's inequality implies that ⁠${\displaystyle L_{x}}$ ⁠ is a bounded linear functional on ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠, and in fact $${\displaystyle |L_{x}(y)|\leq \|x\|_{q}\,\|y\|_{p}}$$ so that the operator norm satisfies $${\displaystyle \|L_{x}\|_{(\ell ^{p})^{*}}\mathrel {\stackrel {\rm {def}}{=}} \sup _{y\in \ell ^{p},y\not =0}{\frac {|L_{x}(y)|}{\|y\|_{p}}}\leq \|x\|_{q}.}$$ In fact, taking ⁠${\displaystyle y}$ ⁠ to be the element of ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ 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 ${\displaystyle L_{x}(y)=\|x\|_{q}}$ , so that in fact $${\displaystyle \|L_{x}\|_{(\ell ^{p})^{*}}=\|x\|_{q}.}$$ Conversely, given a bounded linear functional ⁠${\displaystyle L}$ ⁠ on ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠, the sequence defined by ⁠${\displaystyle x_{n}=L(e_{n})}$ ⁠ lies in ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠. Thus the mapping ⁠${\displaystyle x\mapsto L_{x}}$ ⁠ gives an isometry $${\displaystyle \kappa _{q}:\ell ^{q}\to (\ell ^{p})^{*}.}$$

The map $${\displaystyle \ell ^{q}\xrightarrow {\kappa _{q}} (\ell ^{p})^{*}\xrightarrow {(\kappa _{q}^{*})^{-1}} (\ell ^{q})^{**}}$$ obtained by composing ⁠${\displaystyle \kappa _{p}}$ ⁠ with the inverse of its transpose coincides with the canonical injection of ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠ into its double dual. As a consequence ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠ is a reflexive space. By abuse of notation, it is typical to identify ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠ with the dual of ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠: ⁠${\displaystyle \textstyle (\ell ^{p})^{*}=\ell ^{q}}$ ⁠. Then reflexivity is understood by the sequence of identifications ⁠${\displaystyle \textstyle (\ell ^{p})^{**}=(\ell ^{q})^{*}=\ell ^{p}}$ ⁠.

The space ⁠${\displaystyle c_{0}}$ ⁠ is defined as the space of all sequences converging to zero, with norm identical to ${\displaystyle \|x\|_{\infty }}$ . It is a closed subspace of ⁠${\displaystyle \textstyle \ell ^{\infty }}$ ⁠, hence a Banach space. The dual of ⁠${\displaystyle c_{0}}$ ⁠ is ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠; the dual of ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠ is ⁠${\displaystyle \textstyle \ell ^{\infty }}$ ⁠. For the case of natural numbers index set, the ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ and ⁠${\displaystyle c_{0}}$ ⁠ are separable, with the sole exception of ⁠${\displaystyle \textstyle \ell ^{\infty }}$ ⁠. The dual of ⁠${\displaystyle \textstyle \ell ^{\infty }}$ ⁠ is the ba space.

The spaces ⁠${\displaystyle c_{0}}$ ⁠ and ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ (for ⁠${\displaystyle 1\leq p<\infty }$ ⁠) have a canonical unconditional Schauder basis ⁠${\displaystyle \{e_{i}:i=1,2,\ldots \}}$ ⁠, where ⁠${\displaystyle e_{i}}$ ⁠ is the sequence which is zero but for a ⁠${\displaystyle 1}$ ⁠ in the ⁠${\displaystyle i}$ ⁠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 ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ or of ⁠${\displaystyle c_{0}}$ ⁠, 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 ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠, was answered in the affirmative by Banach & Mazur (1933). That is, for every separable Banach space ⁠${\displaystyle X}$ ⁠, there exists a quotient map ⁠${\displaystyle \textstyle Q:\ell ^{1}\to X}$ ⁠, so that ⁠${\displaystyle X}$ ⁠ is isomorphic to ⁠${\displaystyle \textstyle \ell ^{1}/\ker Q}$ ⁠. In general, ⁠${\displaystyle \operatorname {ker} Q}$ ⁠ is not complemented in ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠, that is, there does not exist a subspace ⁠${\displaystyle Y}$ ⁠ of ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠ such that ⁠${\displaystyle \textstyle \ell ^{1}=Y\oplus \ker Q}$ ⁠. In fact, ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ⁠${\displaystyle \textstyle X=\ell ^{p}}$ ⁠; since there are uncountably many such ⁠${\displaystyle X}$ ⁠'s, and since no ⁠${\displaystyle \textstyle \ell ^{p}}$ ⁠ is isomorphic to any other, there are thus uncountably many ker Q's).

Except for the trivial finite-dimensional case, an unusual feature of ⁠${\displaystyle \textstyle \ell ^{q}}$ ⁠ is that it is not polynomially reflexive.

ℓ^p spaces are increasing in p

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

ℓ² is isomorphic to all separable, infinite dimensional Hilbert spaces

Let ⁠${\displaystyle H}$ ⁠ be a separable Hilbert space. Every orthogonal set in ⁠${\displaystyle H}$ ⁠ is at most countable (i.e. has finite dimension or ⁠${\displaystyle \aleph _{0}}$ ⁠).^[2] The following two items are related:

• If ⁠${\displaystyle H}$ ⁠ is infinite dimensional, then it is isomorphic to ⁠${\displaystyle \textstyle \ell ^{2}}$ ⁠,
• If ⁠${\displaystyle \operatorname {dim} (H)=N}$ ⁠, then ⁠${\displaystyle H}$ ⁠ is isomorphic to ⁠${\displaystyle \textstyle \mathbb {C} ^{N}}$ ⁠.

Properties of ℓ¹ spaces

A sequence of elements in ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠ converges in the space of complex sequences ⁠${\displaystyle \textstyle \ell ^{1}}$ ⁠ if and only if it converges weakly in this space.^[3] If ⁠${\displaystyle K}$ ⁠ is a subset of this space, then the following are equivalent:^[3]

1. ⁠${\displaystyle K}$ ⁠ is compact;
2. ⁠${\displaystyle K}$ ⁠ is weakly compact;
3. ⁠${\displaystyle K}$ ⁠ is bounded, closed, and equismall at infinity.

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

See also

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

References

1. Jarchow 1981, pp. 129–130.
2. Debnath, Lokenath; Mikusinski, Piotr (2005). Hilbert Spaces with Applications. Elsevier. pp. 120–121. ISBN 978-0-12-2084386.
3. Trèves 2006, pp. 451–458.

Bibliography

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