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.

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

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:

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.

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 $${\displaystyle d(x,y)~=~\sum _{n}\left|x_{n}-y_{n}\right|^{p}.}$$

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 ${\displaystyle c_{00}.}$

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.

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.

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.

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}$.

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 ⁠${\displaystyle \textstyle \mathbb {C} ^{N}}$⁠.

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 ⁠${\displaystyle \textstyle s=\left(s_{n}\right)_{n=1}^{\infty }\in K}$⁠.

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

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• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.
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• Pitt, H.R. (1936), "A note on bilinear forms", J. London Math. Soc., 11 (3): 174–180, doi:10.1112/jlms/s1-11.3.174.
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• Schur, J. (1921), "Über lineare Transformationen in der Theorie der unendlichen Reihen", Journal für die reine und angewandte Mathematik, 151: 79–111, doi:10.1515/crll.1921.151.79.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.