Infunctional analysis and related areas ofmathematics, asequence space is avector space whose elements are infinitesequences ofreal orcomplex numbers. Equivalently, it is afunction space whose elements are functions from thenatural numbers to thefield⁠${\displaystyle \mathbb{K}}$⁠ of real or complex numbers. The set of all such functions is naturally identified with the set of all possibleinfinite sequences with elements in⁠${\displaystyle \mathbb{K}}$⁠, and can be turned into avector space under the operations ofpointwise addition of functions and pointwise scalar multiplication. All sequence spaces arelinear subspaces of this space. Sequence spaces are typically equipped with anorm, or at least the structure of atopological vector space.

The most important sequence spaces in analysis are the⁠${\displaystyle {\ell }^p}$⁠ spaces, consisting of the⁠${\displaystyle p}$⁠-power summable sequences, with the⁠${\displaystyle p}$⁠-norm. These are special cases of⁠${\displaystyle L^p}$⁠ spaces for thecounting measure on the set of natural numbers. Other important classes of sequences likeconvergent sequences ornull sequences form sequence spaces, respectively denoted⁠${\displaystyle c}$⁠ and⁠${\displaystyle c_0}$⁠, with thesup norm. Any sequence space can also be equipped with thetopology ofpointwise convergence, under which it becomes a special kind ofFréchet space calledFK-space.

Asequence${\displaystyle x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}}$ 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${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ of allsequences of elements of⁠${\displaystyle \mathbb{K}}$⁠ is avector space forcomponentwise 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 componentwisescalar multiplication$${\displaystyle \alpha {\left(x_n\right)}_{n\in \mathbb{N}}={\left(\alpha x_n\right)}_{n\in \mathbb{N}}.}$$

Asequence space is anylinear subspace of⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠.

As a topological space,⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ is naturally endowed with theproduct topology. Under this topology,⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ isFréchet, meaning that it is acomplete,metrizable,locally convextopological vector space (TVS). However, this topology is rather pathological: there are nocontinuous norms on⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ (and thus the product topology cannotbe defined by anynorm).^[1] Among Fréchet spaces,⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ is minimal in having no continuous norms:

But the product topology is also unavoidable:⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ does not admit astrictly coarser Hausdorff, locally convex topology.^[1] For that reason, the study of sequences begins by finding a strictlinear subspace of interest, and endowing it with a topologydifferent from thesubspace topology.

For⁠⁠,⁠${\displaystyle {\ell }^p}$⁠ is the subspace of⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ consisting of all sequences${\displaystyle x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}}$ satisfying

If⁠${\displaystyle p\ge 1}$⁠, then the real-valued function${\displaystyle \Vert \cdot {\Vert }_p}$ on⁠${\displaystyle {\ell }^p}$⁠ defined by$${\displaystyle \Vert x{\Vert }_p=(\sum _n|x_n{|}^p{)}^{1/p}\text{ for all }x\in {\ell }^p}$$defines anorm on⁠${\displaystyle {\ell }^p}$⁠. In fact,⁠${\displaystyle {\ell }^p}$⁠ is acomplete metric space with respect to this norm, and therefore is aBanach space.

If⁠${\displaystyle p=2}$⁠ then⁠${\displaystyle {\ell }^2}$⁠ is also aHilbert space when endowed with its canonicalinner product, called theEuclidean inner product, defined for all⁠${\displaystyle x_{\bullet },y_{\bullet }\in {\ell }^p}$⁠ by$${\displaystyle ⟨x_{\bullet },y_{\bullet }⟩=\sum _n\overline{x_n}{y}_n.}$$The canonical norm induced by this inner product is the usual⁠${\displaystyle {\ell }^2}$⁠-norm, meaning that${\displaystyle \Vert \mathbf{x}{\Vert }_2=\sqrt{⟨\mathbf{x},\mathbf{x}⟩}}$ for all⁠${\displaystyle \mathbf{x}\in {\ell }^p}$⁠.

If⁠${\displaystyle p=\mathrm{\infty }}$⁠, then⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠ is defined to be the space of allbounded sequences endowed with the norm$${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}=\underset{n}{sup}|x_n|,}$$⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠ is also a Banach space.

If⁠⁠, then⁠${\displaystyle {\ell }^p}$⁠ does not carry a norm, but rather ametric defined by$${\displaystyle d(x,y)=\sum _n{\left|x_n-y_n\right|}^p.}$$

Aconvergent sequence is any sequence${\displaystyle x_{\bullet }\in {\mathbb{K}}^{\mathbb{N}}}$ such that${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{x}_n}$ exists. The set⁠${\displaystyle c}$⁠ of all convergent sequences is a vector subspace of⁠⁠ called thespace of convergent sequences. Since every convergent sequence is bounded,⁠${\displaystyle c}$⁠ is a linear subspace of⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠. Moreover, this sequence space is a closed subspace of⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠ with respect to thesupremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to⁠${\displaystyle 0}$⁠ is called anull sequence and is said tovanish. The set of all sequences that converge to⁠${\displaystyle 0}$⁠ is a closed vector subspace of⁠${\displaystyle c}$⁠ that when endowed with thesupremum norm becomes a Banach space that is denoted by⁠${\displaystyle c_0}$⁠ and is called thespace of null sequences or thespace of vanishing sequences.

Thespace 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${\displaystyle (x_{nk}{)}_{k\in \mathbb{N}}}$ where${\displaystyle x_{nk}=1/k}$ for the first${\displaystyle n}$ entries (for${\displaystyle k=1,\dots ,n}$) and is zero everywhere else (that is,${\displaystyle (x_{nk}{)}_{k\in \mathbb{N}}=}$${\displaystyle (1,\frac{1}{2},\dots ,}$${\displaystyle \frac{1}{n-1},\frac{1}{n},}$${\displaystyle 0,0,\dots )}$) is aCauchy sequence but it does not converge to a sequence in${\displaystyle c_{00}.}$

Let$${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}=\left\{\left(x_1,x_2,\dots \right)\in {\mathbb{K}}^{\mathbb{N}}:\text{all but finitely many }{x}_i\text{ equal }0\right\}}$$

denote thespace of finite sequences over⁠${\displaystyle \mathbb{K}}$⁠. As a vector space,${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$ is equal to⁠${\displaystyle c_{00}}$⁠, but⁠${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$⁠ has a different topology.

For everynatural number⁠${\displaystyle n\in \mathbb{N}}$⁠, let⁠${\displaystyle {\mathbb{K}}^n}$⁠ denote the usualEuclidean space endowed with theEuclidean topology and let${\displaystyle {\mathrm{In}}_{{\mathbb{K}}^n}:{\mathbb{K}}^n\to {\mathbb{K}}^{\mathrm{\infty }}}$ denote the canonical inclusion$${\displaystyle {\mathrm{In}}_{{\mathbb{K}}^n}\left(x_1,\dots ,x_n\right)=\left(x_1,\dots ,x_n,0,0,\dots \right).}$$Theimage of each inclusion is$${\displaystyle \mathrm{Im}\left({\mathrm{In}}_{{\mathbb{K}}^n}\right)=\left\{\left(x_1,\dots ,x_n,0,0,\dots \right):x_1,\dots ,x_n\in \mathbb{K}\right\}={\mathbb{K}}^n\times \left\{(0,0,\dots )\right\}}$$and consequently,$${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}=\bigcup _{n\in \mathbb{N}}\mathrm{Im}\left({\mathrm{In}}_{{\mathbb{K}}^n}\right).}$$

This family of inclusions gives⁠${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$⁠ afinal topology⁠${\displaystyle {\tau }^{\mathrm{\infty }}}$⁠, defined to be thefinest topology on⁠${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$⁠ such that all the inclusions are continuous (an example of acoherent topology). With this topology,⁠${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$⁠ becomes acomplete,Hausdorff,locally convex,sequential,topological vector space that isnotFréchet–Urysohn. The topology⁠${\displaystyle {\tau }^{\mathrm{\infty }}}$⁠ is alsostrictly finer than thesubspace topology induced on⁠${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$⁠ by⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠.

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

Often, each image${\displaystyle \mathrm{Im}\left({\mathrm{In}}_{{\mathbb{K}}^n}\right)}$ is identified with the corresponding⁠${\displaystyle {\mathbb{K}}^n}$⁠; explicitly, the elements${\displaystyle \left(x_1,\dots ,x_n\right)\in {\mathbb{K}}^n}$ and${\displaystyle \left(x_1,\dots ,x_n,0,0,0,\dots \right)}$ are identified. This is facilitated by the fact that the subspace topology on${\displaystyle \mathrm{Im}\left({\mathrm{In}}_{{\mathbb{K}}^n}\right)}$, thequotient topology from the map${\displaystyle {\mathrm{In}}_{{\mathbb{K}}^n}}$, and the Euclidean topology on⁠${\displaystyle {\mathbb{K}}^n}$⁠ all coincide. With this identification,${\displaystyle \left(\left({\mathbb{K}}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }}\right),{\left({\mathrm{In}}_{{\mathbb{K}}^n}\right)}_{n\in \mathbb{N}}\right)}$ is thedirect limit of the directed system${\displaystyle \left({\left({\mathbb{K}}^n\right)}_{n\in \mathbb{N}},{\left({\mathrm{In}}_{{\mathbb{K}}^m\to {\mathbb{K}}^n}\right)}_{m\le n\in \mathbb{N}},\mathbb{N}\right),}$ where every inclusion adds trailing zeros:$${\displaystyle {\mathrm{In}}_{{\mathbb{K}}^m\to {\mathbb{K}}^n}\left(x_1,\dots ,x_m\right)=\left(x_1,\dots ,x_m,0,\dots ,0\right).}$$ This shows${\displaystyle \left({\mathbb{K}}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }}\right)}$ is anLB-space.

The space of boundedseries, denote bybs, is the space of sequences⁠${\displaystyle x}$⁠ for which

This space, when equipped with the norm$${\displaystyle \Vert x{\Vert }_{bs}=\underset{n}{sup}|\sum _{i=0}^n{x}_i|,}$$

is a Banach space isometrically isomorphic to${\displaystyle {\ell }^{\mathrm{\infty }},}$ via thelinear mapping$${\displaystyle (x_n{)}_{n\in \mathbb{N}}\mapsto (\sum _{i=0}^n{x}_i{)}_{n\in \mathbb{N}}.}$$

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${\displaystyle \mathrm{\Phi }}$⁠ or${\displaystyle c_{00}}$ is defined to be the space of all infinite sequences with only a finite number of non-zero terms (sequences withfinite support). This set isdense in many sequence spaces.

The space⁠${\displaystyle {\ell }^2}$⁠ is the only⁠${\displaystyle {\ell }^p}$⁠ space that is aHilbert space, since any norm that is induced by aninner product should satisfy theparallelogram law

$${\displaystyle \Vert x+y{\Vert }_p^2+\Vert x-y{\Vert }_p^2=2\Vert x{\Vert }_p^2+2\Vert y{\Vert }_p^2.}$$

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

Each⁠${\displaystyle {\ell }^p}$⁠ is distinct, in that⁠${\displaystyle {\ell }^p}$⁠ is a strictsubset of⁠${\displaystyle {\ell }^s}$⁠ whenever⁠⁠; furthermore,⁠${\displaystyle {\ell }^p}$⁠ is not linearlyisomorphic to⁠${\displaystyle {\ell }^s}$⁠ when⁠${\displaystyle p\ne s}$⁠. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from⁠${\displaystyle {\ell }^s}$⁠ to⁠${\displaystyle {\ell }^p}$⁠ iscompact when⁠⁠. 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 bestrictly singular.

If⁠⁠, then the(continuous) dual space of⁠${\displaystyle {\ell }^p}$⁠ is isometrically isomorphic to⁠${\displaystyle {\ell }^q}$⁠, where⁠${\displaystyle q}$⁠ is theHölder conjugate of⁠${\displaystyle p}$⁠:⁠${\displaystyle 1/p+1/q=1}$⁠. The specific isomorphism associates to an element⁠${\displaystyle x}$⁠ of⁠${\displaystyle {\ell }^q}$⁠ the functional$${\displaystyle L_x(y)=\sum _n{x}_n{y}_n}$$for⁠${\displaystyle y}$⁠ in⁠${\displaystyle {\ell }^p}$⁠.Hölder's inequality implies that⁠${\displaystyle L_x}$⁠ is a bounded linear functional on⁠${\displaystyle {\ell }^p}$⁠, and in fact$${\displaystyle |L_x(y)|\le \Vert x{\Vert }_q\Vert y{\Vert }_p}$$so that the operator norm satisfies$${\displaystyle \Vert L_x{\Vert }_{({\ell }^p{)}^{\ast }}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{y\in {\ell }^p,y\ne 0}{sup}\frac{|L_x(y)|}{\Vert y{\Vert }_p}\le \Vert x{\Vert }_q.}$$In fact, taking⁠${\displaystyle y}$⁠ to be the element of⁠${\displaystyle {\ell }^p}$⁠ withgives${\displaystyle L_x(y)=\Vert x{\Vert }_q}$, so that in fact$${\displaystyle \Vert L_x{\Vert }_{({\ell }^p{)}^{\ast }}=\Vert x{\Vert }_q.}$$Conversely, given a bounded linear functional⁠${\displaystyle L}$⁠ on⁠${\displaystyle {\ell }^p}$⁠, the sequence defined by⁠${\displaystyle x_n=L(e_n)}$⁠ lies in⁠${\displaystyle {\ell }^q}$⁠. Thus the mapping⁠${\displaystyle x\mapsto L_x}$⁠ gives an isometry$${\displaystyle {\kappa }_q:{\ell }^q\to ({\ell }^p{)}^{\ast }.}$$

The map$${\displaystyle {\ell }^q\stackrel{{\kappa }_q}{\to }({\ell }^p{)}^{\ast }\stackrel{({\kappa }_q^{\ast }{)}^{-1}}{\to }({\ell }^q{)}^{\ast \ast }}$$obtained by composing⁠${\displaystyle {\kappa }_p}$⁠ with the inverse of itstranspose coincides with thecanonical injection of⁠${\displaystyle {\ell }^q}$⁠ into itsdouble dual. As a consequence⁠${\displaystyle {\ell }^q}$⁠ is areflexive space. Byabuse of notation, it is typical to identify⁠${\displaystyle {\ell }^q}$⁠ with the dual of⁠${\displaystyle {\ell }^p}$⁠:⁠${\displaystyle ({\ell }^p{)}^{\ast }={\ell }^q}$⁠. Then reflexivity is understood by the sequence of identifications⁠${\displaystyle ({\ell }^p{)}^{\ast \ast }=({\ell }^q{)}^{\ast }={\ell }^p}$⁠.

The space⁠${\displaystyle c_0}$⁠ is defined as the space of all sequences converging to zero, with norm identical to${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}}$. It is a closed subspace of⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠, hence a Banach space. Thedual of⁠${\displaystyle c_0}$⁠ is⁠${\displaystyle {\ell }^1}$⁠; the dual of⁠${\displaystyle {\ell }^1}$⁠ is⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠. For the case of natural numbers index set, the⁠${\displaystyle {\ell }^p}$⁠ and⁠${\displaystyle c_0}$⁠ areseparable, with the sole exception of⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠. The dual of⁠${\displaystyle {\ell }^{\mathrm{\infty }}}$⁠ is theba space.

The spaces⁠${\displaystyle c_0}$⁠ and⁠${\displaystyle {\ell }^p}$⁠ (for⁠⁠) have a canonical unconditionalSchauder basis⁠${\displaystyle \{e_i:i=1,2,\dots \}}$⁠, 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 theSchur property: In ℓ¹, any sequence that isweakly convergent is alsostrongly convergent (Schur 1921). However, since theweak topology on infinite-dimensional spaces is strictly weaker than thestrong topology, there arenets in ℓ¹ that are weak convergent but not strong convergent.

The⁠${\displaystyle {\ell }^p}$⁠ spaces can beembedded into manyBanach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some⁠${\displaystyle {\ell }^p}$⁠ or of⁠${\displaystyle c_0}$⁠, was answered negatively byB. S. Tsirelson's construction ofTsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to aquotient space of⁠${\displaystyle {\ell }^1}$⁠, was answered in the affirmative byBanach & Mazur (1933). That is, for every separable Banach space⁠${\displaystyle X}$⁠, there exists a quotient map⁠${\displaystyle Q:{\ell }^1\to X}$⁠, so that⁠${\displaystyle X}$⁠ is isomorphic to⁠${\displaystyle {\ell }^1/\ker Q}$⁠. In general,⁠${\displaystyle \ker Q}$⁠ is not complemented in⁠${\displaystyle {\ell }^1}$⁠, that is, there does not exist a subspace⁠${\displaystyle Y}$⁠ of⁠${\displaystyle {\ell }^1}$⁠ such that⁠${\displaystyle {\ell }^1=Y\oplus \ker Q}$⁠. In fact,⁠${\displaystyle {\ell }^1}$⁠ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take⁠${\displaystyle X={\ell }^p}$⁠; since there are uncountably many such⁠${\displaystyle X}$⁠'s, and since no⁠${\displaystyle {\ell }^p}$⁠ is isomorphic to any other, there are thus uncountably many kerQ's).

Except for the trivial finite-dimensional case, an unusual feature of⁠${\displaystyle {\ell }^q}$⁠ is that it is notpolynomially reflexive.

For⁠${\displaystyle p\in [1,\mathrm{\infty }]}$⁠, the spaces⁠${\displaystyle {\ell }^p}$⁠ are increasing in⁠${\displaystyle p}$⁠, with the inclusion operator being continuous: for⁠⁠, one has${\displaystyle \Vert x{\Vert }_q\le \Vert x{\Vert }_p}$. Indeed, the inequality is homogeneous in the⁠${\displaystyle x_i}$⁠, so it is sufficient to prove it under the assumption that${\displaystyle \Vert x{\Vert }_p=1}$. In this case, we need only show that${\displaystyle \sum |x_i{|}^q\le 1}$ for⁠${\displaystyle q>p}$⁠. But if${\displaystyle \Vert x{\Vert }_p=1}$, then${\displaystyle |x_i|\le 1}$ for all⁠${\displaystyle i}$⁠, and then${\displaystyle \sum |x_i{|}^q\le }$${\displaystyle \sum |x_i{|}^p=1}$.

Let⁠${\displaystyle H}$⁠ be aseparable Hilbert space. Every orthogonal set in⁠${\displaystyle H}$⁠ is at mostcountable (i.e. has finitedimension or⁠${\displaystyle {\mathrm{\aleph }}_0}$⁠).^[2] The following two items are related:

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

A sequence of elements in⁠${\displaystyle {\ell }^1}$⁠ converges in the space of complex sequences⁠${\displaystyle {\ell }^1}$⁠ 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${\displaystyle K}$⁠ beingequismall at infinity means that for every⁠${\displaystyle \epsilon >0}$⁠, there exists a natural number${\displaystyle n_{\epsilon }\ge 0}$ such that for all⁠${\displaystyle s={\left(s_n\right)}_{n=1}^{\mathrm{\infty }}\in K}$⁠.

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

• Banach, Stefan; Mazur, S. (1933), "Zur Theorie der linearen Dimension",Studia Mathematica,4:100–112,doi:10.4064/sm-4-1-100-112.
• Dunford, Nelson; Schwartz, Jacob T. (1958),Linear operators, volume I, Wiley-Interscience.
• Jarchow, Hans (1981).Locally convex spaces. Stuttgart: B.G. Teubner.ISBN 978-3-519-02224-4.OCLC 8210342.
• 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.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
• 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.

• Sequence spaces
• Functional analysis
• Sequences and series

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