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 {\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 ⁠K {\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 ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ spaces, consisting of the ⁠p {\displaystyle p}⁠-power summable sequences, with the ⁠p {\displaystyle p}⁠-norm. These are special cases of ⁠L p {\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 ⁠c {\displaystyle c}⁠ and ⁠c 0 {\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 x ∙ = ( x n ) n ∈ N {\displaystyle \textstyle x_{\bullet }=(x_{n})_{n\in \mathbb {N} }} in a set ⁠X {\displaystyle X}⁠ is an ⁠X {\displaystyle X}⁠-valued map x ∙ : N → X {\displaystyle x_{\bullet }:\mathbb {N} \to X} whose value at ⁠n ∈ N {\displaystyle n\in \mathbb {N} }⁠ is denoted by ⁠x n {\displaystyle x_{n}}⁠ instead of the usual parentheses notation ⁠x ( n ) {\displaystyle x(n)}⁠.

Space of all sequences

Let ⁠K {\displaystyle \mathbb {K} }⁠ denote the field either of real or complex numbers. The set ⁠K N {\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}⁠ of all sequences of elements of ⁠K {\displaystyle \mathbb {K} }⁠ is a vector space for componentwise addition ( x n ) n ∈ N + ( y n ) n ∈ N = ( x n + y n ) n ∈ N , {\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 α ( x n ) n ∈ N = ( α x n ) n ∈ N . {\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 ⁠K N {\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}⁠.

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

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

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

But the product topology is also unavoidable: ⁠K N {\displaystyle \textstyle \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

For ⁠0 < p < ∞ {\displaystyle 0<p<\infty }⁠, ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ is the subspace of ⁠K N {\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }}⁠ consisting of all sequences x ∙ = ( x n ) n ∈ N {\displaystyle \textstyle x_{\bullet }=(x_{n})_{n\in \mathbb {N} }} satisfying ∑ n | x n | p < ∞ . {\displaystyle \sum _{n}|x_{n}|^{p}<\infty .}

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

If ⁠p = 2 {\displaystyle p=2}⁠ then ⁠ℓ 2 {\displaystyle \textstyle \ell ^{2}}⁠ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all ⁠x ∙ , y ∙ ∈ ℓ p {\displaystyle \textstyle x_{\bullet },y_{\bullet }\in \ell ^{p}}⁠ by ⟨ x ∙ , y ∙ ⟩ = ∑ n x n ¯ y n . {\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 ⁠ℓ 2 {\displaystyle \textstyle \ell ^{2}}⁠-norm, meaning that ‖ x ‖ 2 = ⟨ x , x ⟩ {\displaystyle \textstyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} for all ⁠x ∈ ℓ p {\displaystyle \textstyle \mathbf {x} \in \ell ^{p}}⁠.

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

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

c , c 0 and c 00

A convergent sequence is any sequence x ∙ ∈ K N {\displaystyle \textstyle x_{\bullet }\in \mathbb {K} ^{\mathbb {N} }} such that lim n → ∞ x n {\displaystyle \textstyle \lim _{n\to \infty }x_{n}} exists. The set ⁠c {\displaystyle c}⁠ of all convergent sequences is a vector subspace of ⁠K N < {\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }<}⁠ called the space of convergent sequences. Since every convergent sequence is bounded, ⁠c {\displaystyle c}⁠ is a linear subspace of ⁠ℓ ∞ {\displaystyle \ell ^{\infty }}⁠. Moreover, this sequence space is a closed subspace of ⁠ℓ ∞ {\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 ⁠0 {\displaystyle 0}⁠ is called a null sequence and is said to vanish. The set of all sequences that converge to ⁠0 {\displaystyle 0}⁠ is a closed vector subspace of ⁠c {\displaystyle c}⁠ that when endowed with the supremum norm becomes a Banach space that is denoted by ⁠c 0 {\displaystyle c_{0}}⁠ and is called the space of null sequences or the space of vanishing sequences.

The space of eventually zero sequences, ⁠c 00 {\displaystyle c_{00}}⁠, is the subspace of ⁠c 0 {\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 ( x n k ) k ∈ N {\displaystyle \textstyle (x_{nk})_{k\in \mathbb {N} }} where x n k = 1 / k {\displaystyle x_{nk}=1/k} for the first n {\displaystyle n} entries (for k = 1 , … , n {\displaystyle k=1,\ldots ,n}) and is zero everywhere else (that is, ( x n k ) k ∈ N = {\displaystyle \textstyle (x_{nk})_{k\in \mathbb {N} }={}\!}( 1 , 1 2 , … , {\displaystyle {\bigl (}1,{\tfrac {1}{2}},\ldots ,{}}1 n − 1 , 1 n , {\displaystyle {\tfrac {1}{n-1}},{\tfrac {1}{n}},{}}0 , 0 , … ) {\displaystyle 0,0,\ldots {\bigr )}}) is a Cauchy sequence but it does not converge to a sequence in c 00 . {\displaystyle c_{00}.}

Space of all finite sequences

Let K ∞ = { ( x 1 , x 2 , … ) ∈ K N : all but finitely many x i equal 0 } {\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 ⁠K {\displaystyle \mathbb {K} }⁠. As a vector space, K ∞ {\displaystyle \textstyle \mathbb {K} ^{\infty }} is equal to ⁠c 00 {\displaystyle c_{00}}⁠, but ⁠K ∞ {\displaystyle \textstyle \mathbb {K} ^{\infty }}⁠ has a different topology.

For every natural number ⁠n ∈ N {\displaystyle n\in \mathbb {N} }⁠, let ⁠K n {\displaystyle \textstyle \mathbb {K} ^{n}}⁠ denote the usual Euclidean space endowed with the Euclidean topology and let In K n : K n → K ∞ {\displaystyle \textstyle \operatorname {In} _{\mathbb {K} ^{n}}:\mathbb {K} ^{n}\to \mathbb {K} ^{\infty }} denote the canonical inclusion In K n ⁡ ( x 1 , … , x n ) = ( x 1 , … , x n , 0 , 0 , … ) . {\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 Im ⁡ ( In K n ) = { ( x 1 , … , x n , 0 , 0 , … ) : x 1 , … , x n ∈ K } = K n × { ( 0 , 0 , … ) } {\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, K ∞ = ⋃ n ∈ N Im ⁡ ( In K n ) . {\displaystyle \mathbb {K} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right).}

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

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

Often, each image Im ⁡ ( In K n ) {\displaystyle \textstyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right)} is identified with the corresponding ⁠K n {\displaystyle \textstyle \mathbb {K} ^{n}}⁠; explicitly, the elements ( x 1 , … , x n ) ∈ K n {\displaystyle \textstyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {K} ^{n}} and ( x 1 , … , x n , 0 , 0 , 0 , … ) {\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 Im ⁡ ( In K n ) {\displaystyle \textstyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {K} ^{n}}\right)}, the quotient topology from the map In K n {\displaystyle \textstyle \operatorname {In} _{\mathbb {K} ^{n}}}, and the Euclidean topology on ⁠K n {\displaystyle \textstyle \mathbb {K} ^{n}}⁠ all coincide. With this identification, ( ( K ∞ , τ ∞ ) , ( In K n ) n ∈ N ) {\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 ( ( K n ) n ∈ N , ( In K m → K n ) m ≤ n ∈ N , N ) , {\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: In K m → K n ⁡ ( x 1 , … , x m ) = ( x 1 , … , x m , 0 , … , 0 ) . {\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 ( K ∞ , τ ∞ ) {\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 ⁠x {\displaystyle x}⁠ for which sup n | ∑ i = 0 n x i | < ∞ . {\displaystyle \sup _{n}{\biggl \vert }\sum _{i=0}^{n}x_{i}{\biggr \vert }<\infty .}

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

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

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

The space ⁠Φ {\displaystyle \Phi }⁠ or c 00 {\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 c 0

The space ⁠ℓ 2 {\displaystyle \textstyle \ell ^{2}}⁠ is the only ⁠ℓ p {\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

‖ x + y ‖ p 2 + ‖ x − y ‖ p 2 = 2 ‖ x ‖ p 2 + 2 ‖ y ‖ p 2 . {\displaystyle \|x+y\|_{p}^{2}+\|x-y\|_{p}^{2}=2\|x\|_{p}^{2}+2\|y\|_{p}^{2}.}

Substituting two distinct unit vectors for ⁠x {\displaystyle x}⁠ and ⁠y {\displaystyle y}⁠ directly shows that the identity is not true unless ⁠p = 2 {\displaystyle p=2}⁠.

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

If ⁠1 < p < ∞ {\displaystyle 1<p<\infty }⁠, then the (continuous) dual space of ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ is isometrically isomorphic to ⁠ℓ q {\displaystyle \textstyle \ell ^{q}}⁠, where ⁠q {\displaystyle q}⁠ is the Hölder conjugate of ⁠p {\displaystyle p}⁠: ⁠1 / p + 1 / q = 1 {\displaystyle 1/p+1/q=1}⁠. The specific isomorphism associates to an element ⁠x {\displaystyle x}⁠ of ⁠ℓ q {\displaystyle \textstyle \ell ^{q}}⁠ the functional L x ( y ) = ∑ n x n y n {\displaystyle L_{x}(y)=\sum _{n}x_{n}y_{n}} for ⁠y {\displaystyle y}⁠ in ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠. Hölder's inequality implies that ⁠L x {\displaystyle L_{x}}⁠ is a bounded linear functional on ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠, and in fact | L x ( y ) | ≤ ‖ x ‖ q ‖ y ‖ p {\displaystyle |L_{x}(y)|\leq \|x\|_{q}\,\|y\|_{p}} so that the operator norm satisfies ‖ L x ‖ ( ℓ p ) ∗ = d e f sup y ∈ ℓ p , y ≠ 0 | L x ( y ) | ‖ y ‖ p ≤ ‖ x ‖ q . {\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 ⁠y {\displaystyle y}⁠ to be the element of ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ with y n = { 0 if x n = 0 x n − 1 | x n | q if x n ≠ 0 {\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 L x ( y ) = ‖ x ‖ q {\displaystyle L_{x}(y)=\|x\|_{q}}, so that in fact ‖ L x ‖ ( ℓ p ) ∗ = ‖ x ‖ q . {\displaystyle \|L_{x}\|_{(\ell ^{p})^{*}}=\|x\|_{q}.} Conversely, given a bounded linear functional ⁠L {\displaystyle L}⁠ on ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠, the sequence defined by ⁠x n = L ( e n ) {\displaystyle x_{n}=L(e_{n})}⁠ lies in ⁠ℓ q {\displaystyle \textstyle \ell ^{q}}⁠. Thus the mapping ⁠x ↦ L x {\displaystyle x\mapsto L_{x}}⁠ gives an isometry κ q : ℓ q → ( ℓ p ) ∗ . {\displaystyle \kappa _{q}:\ell ^{q}\to (\ell ^{p})^{*}.}

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

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

The spaces ⁠c 0 {\displaystyle c_{0}}⁠ and ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ (for ⁠1 ≤ p < ∞ {\displaystyle 1\leq p<\infty }⁠) have a canonical unconditional Schauder basis ⁠{ e i : i = 1 , 2 , … } {\displaystyle \{e_{i}:i=1,2,\ldots \}}⁠, where ⁠e i {\displaystyle e_{i}}⁠ is the sequence which is zero but for a ⁠1 {\displaystyle 1}⁠ in the ⁠i {\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 ⁠ℓ p {\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 ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ or of ⁠c 0 {\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 ⁠ℓ 1 {\displaystyle \textstyle \ell ^{1}}⁠, was answered in the affirmative by Banach & Mazur (1933). That is, for every separable Banach space ⁠X {\displaystyle X}⁠, there exists a quotient map ⁠Q : ℓ 1 → X {\displaystyle \textstyle Q:\ell ^{1}\to X}⁠, so that ⁠X {\displaystyle X}⁠ is isomorphic to ⁠ℓ 1 / ker ⁡ Q {\displaystyle \textstyle \ell ^{1}/\ker Q}⁠. In general, ⁠ker ⁡ Q {\displaystyle \operatorname {ker} Q}⁠ is not complemented in ⁠ℓ 1 {\displaystyle \textstyle \ell ^{1}}⁠, that is, there does not exist a subspace ⁠Y {\displaystyle Y}⁠ of ⁠ℓ 1 {\displaystyle \textstyle \ell ^{1}}⁠ such that ⁠ℓ 1 = Y ⊕ ker ⁡ Q {\displaystyle \textstyle \ell ^{1}=Y\oplus \ker Q}⁠. In fact, ⁠ℓ 1 {\displaystyle \textstyle \ell ^{1}}⁠ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ⁠X = ℓ p {\displaystyle \textstyle X=\ell ^{p}}⁠; since there are uncountably many such ⁠X {\displaystyle X}⁠'s, and since no ⁠ℓ p {\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 ⁠ℓ q {\displaystyle \textstyle \ell ^{q}}⁠ is that it is not polynomially reflexive.

ℓ p spaces are increasing in p

For ⁠p ∈ [ 1 , ∞ ] {\displaystyle p\in [1,\infty ]}⁠, the spaces ⁠ℓ p {\displaystyle \textstyle \ell ^{p}}⁠ are increasing in ⁠p {\displaystyle p}⁠, with the inclusion operator being continuous: for ⁠1 ≤ p < q ≤ ∞ {\displaystyle 1\leq p<q\leq \infty }⁠, one has ‖ x ‖ q ≤ ‖ x ‖ p {\displaystyle \|x\|_{q}\leq \|x\|_{p}}. Indeed, the inequality is homogeneous in the ⁠x i {\displaystyle x_{i}}⁠, so it is sufficient to prove it under the assumption that ‖ x ‖ p = 1 {\displaystyle \|x\|_{p}=1}. In this case, we need only show that ∑ | x i | q ≤ 1 {\displaystyle \textstyle \sum |x_{i}|^{q}\leq 1} for ⁠q > p {\displaystyle q>p}⁠. But if ‖ x ‖ p = 1 {\displaystyle \|x\|_{p}=1}, then | x i | ≤ 1 {\displaystyle |x_{i}|\leq 1} for all ⁠i {\displaystyle i}⁠, and then ∑ | x i | q ≤ {\displaystyle \textstyle \sum |x_{i}|^{q}\leq {}\!}∑ | x i | p = 1 {\displaystyle \textstyle \sum |x_{i}|^{p}=1}.

ℓ 2 is isomorphic to all separable, infinite dimensional Hilbert spaces

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

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

Properties of ℓ 1 spaces

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

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

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

See also

Bibliography

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