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

Space of all sequences

Let ⁠ $𝕂$ ⁠ denote the field either of real or complex numbers. The set ⁠ $𝕂^{ℕ}$ ⁠ of all sequences of elements of ⁠ $𝕂$ ⁠ is a vector space for componentwise addition ${(x_{n})}_{n \in ℕ} + {(y_{n})}_{n \in ℕ} = {(x_{n} + y_{n})}_{n \in ℕ},$ and componentwise scalar multiplication $α {(x_{n})}_{n \in ℕ} = {(α x_{n})}_{n \in ℕ} .$

A sequence space is any linear subspace of ⁠ $𝕂^{ℕ}$ ⁠.

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

Theorem—Let ⁠ $X$ ⁠ be a Fréchet space over ⁠ $𝕂$ ⁠. Then the following are equivalent:

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

But the product topology is also unavoidable: ⁠ $𝕂^{ℕ}$ ⁠ 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.

$ℓ p$ spaces

For ⁠ $0 < p < \infty$ ⁠, ⁠ $ℓ^{p}$ ⁠ is the subspace of ⁠ $𝕂^{ℕ}$ ⁠ consisting of all sequences $x_{∙} = (x_{n})_{n \in ℕ}$ satisfying $\sum_{n} | x_{n} |^{p} < \infty .$

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

If ⁠ $p = 2$ ⁠ then ⁠ $ℓ^{2}$ ⁠ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all ⁠ $x_{∙}, y_{∙} \in ℓ^{p}$ ⁠ by $⟨ x_{∙}, y_{∙} ⟩ = \sum_{n} \overline{x_{n}} y_{n} .$ The canonical norm induced by this inner product is the usual ⁠ $ℓ^{2}$ ⁠-norm, meaning that $‖ 𝐱 ‖_{2} = \sqrt{⟨ 𝐱, 𝐱 ⟩}$ for all ⁠ $𝐱 \in ℓ^{p}$ ⁠.

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

If ⁠ $0 < p < 1$ ⁠, then ⁠ $ℓ^{p}$ ⁠ does not carry a norm, but rather a metric defined by $d (x, y) = \sum_{n} {| x_{n} - y_{n} |}^{p} .$

$c$ , $c 0$ and $c 00$

A convergent sequence is any sequence $x_{∙} \in 𝕂^{ℕ}$ such that ${lim}_{n \to \infty} x_{n}$ exists. The set ⁠ $c$ ⁠ of all convergent sequences is a vector subspace of ⁠ $𝕂^{ℕ}$ ⁠ called the space of convergent sequences. Since every convergent sequence is bounded, ⁠ $c$ ⁠ is a linear subspace of ⁠ $ℓ^{\infty}$ ⁠. Moreover, this sequence space is a closed subspace of ⁠ $ℓ^{\infty}$ ⁠ with respect to the supremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to ⁠ $0$ ⁠ is called a null sequence and is said to vanish. The set of all sequences that converge to ⁠ $0$ ⁠ is a closed vector subspace of ⁠ $c$ ⁠ that when endowed with the supremum norm becomes a Banach space that is denoted by ⁠ $c_{0}$ ⁠ and is called the space of null sequences or the space of vanishing sequences.

The space of eventually zero sequences, ⁠ $c_{00}$ ⁠, is the subspace of ⁠ $c_{0}$ ⁠ 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 $(x_{n k})_{k \in ℕ}$ where $x_{n k} = 1 / k$ for the first $n$ entries (for $k = 1, \dots, n$ ) and is zero everywhere else (that is, $(x_{n k})_{k \in ℕ} =$ $(1, \frac{1}{2}, \dots,$ $\frac{1}{n - 1}, \frac{1}{n},$ $0, 0, \dots)$ ) is a Cauchy sequence but it does not converge to a sequence in $c_{00} .$

Space of all finite sequences

Let $𝕂^{\infty} = {(x_{1}, x_{2}, \dots) \in 𝕂^{ℕ} : all but finitely many x_{i} equal 0}$

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

For every natural number ⁠ $n \in ℕ$ ⁠, let ⁠ $𝕂^{n}$ ⁠ denote the usual Euclidean space endowed with the Euclidean topology and let ${In}_{𝕂^{n}} : 𝕂^{n} \to 𝕂^{\infty}$ denote the canonical inclusion ${In}_{𝕂^{n}} (x_{1}, \dots, x_{n}) = (x_{1}, \dots, x_{n}, 0, 0, \dots) .$ The image of each inclusion is $Im ({In}_{𝕂^{n}}) = {(x_{1}, \dots, x_{n}, 0, 0, \dots) : x_{1}, \dots, x_{n} \in 𝕂} = 𝕂^{n} \times {(0, 0, \dots)}$ and consequently, $𝕂^{\infty} = ⋃_{n \in ℕ} Im ({In}_{𝕂^{n}}) .$

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

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

Often, each image $Im ({In}_{𝕂^{n}})$ is identified with the corresponding ⁠ $𝕂^{n}$ ⁠; explicitly, the elements $(x_{1}, \dots, x_{n}) \in 𝕂^{n}$ and $(x_{1}, \dots, x_{n}, 0, 0, 0, \dots)$ are identified. This is facilitated by the fact that the subspace topology on $Im ({In}_{𝕂^{n}})$ , the quotient topology from the map ${In}_{𝕂^{n}}$ , and the Euclidean topology on ⁠ $𝕂^{n}$ ⁠ all coincide. With this identification, $((𝕂^{\infty}, τ^{\infty}), {({In}_{𝕂^{n}})}_{n \in ℕ})$ is the direct limit of the directed system $({(𝕂^{n})}_{n \in ℕ}, {({In}_{𝕂^{m} \to 𝕂^{n}})}_{m \leq n \in ℕ}, ℕ),$ where every inclusion adds trailing zeros: ${In}_{𝕂^{m} \to 𝕂^{n}} (x_{1}, \dots, x_{m}) = (x_{1}, \dots, x_{m}, 0, \dots, 0) .$ This shows $(𝕂^{\infty}, τ^{\infty})$ is an LB-space.

Other sequence spaces

The space of bounded series, denote by bs, is the space of sequences ⁠ $x$ ⁠ for which $sup_{n} | \sum_{i = 0}^{n} x_{i} | < \infty .$

This space, when equipped with the norm $‖ x ‖_{b s} = sup_{n} | \sum_{i = 0}^{n} x_{i} |,$

is a Banach space isometrically isomorphic to $ℓ^{\infty},$ via the linear mapping $(x_{n})_{n \in ℕ} \mapsto (\sum_{i = 0}^{n} x_{i})_{n \in ℕ} .$

The subspace $c s$ consisting of all convergent series is a subspace that goes over to the space ⁠ $c$ ⁠ under this isomorphism.

The space ⁠ $Φ$ ⁠ 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 $ℓ p$ spaces and the space $c 0$

The space ⁠ $ℓ^{2}$ ⁠ is the only ⁠ $ℓ^{p}$ ⁠ space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law

$‖ x + y ‖_{p}^{2} + ‖ x - y ‖_{p}^{2} = 2 ‖ x ‖_{p}^{2} + 2 ‖ y ‖_{p}^{2} .$

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

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

If ⁠ $1 < p < \infty$ ⁠, then the (continuous) dual space of ⁠ $ℓ^{p}$ ⁠ is isometrically isomorphic to ⁠ $ℓ^{q}$ ⁠, where ⁠ $q$ ⁠ is the Hölder conjugate of ⁠ $p$ ⁠: ⁠ $1 / p + 1 / q = 1$ ⁠. The specific isomorphism associates to an element ⁠ $x$ ⁠ of ⁠ $ℓ^{q}$ ⁠ the functional $L_{x} (y) = \sum_{n} x_{n} y_{n}$ for ⁠ $y$ ⁠ in ⁠ $ℓ^{p}$ ⁠. Hölder's inequality implies that ⁠ $L_{x}$ ⁠ is a bounded linear functional on ⁠ $ℓ^{p}$ ⁠, and in fact $| L_{x} (y) | \leq ‖ x ‖_{q} ‖ y ‖_{p}$ so that the operator norm satisfies $‖ L_{x} ‖_{(ℓ^{p})^{*}} \overset{d e f}{=} sup_{y \in ℓ^{p}, y ≠ 0} \frac{| L_{x} (y) |}{‖ y ‖_{p}} \leq ‖ x ‖_{q} .$ In fact, taking ⁠ $y$ ⁠ to be the element of ⁠ $ℓ^{p}$ ⁠ with $y_{n} = {\begin{matrix} 0 & if x_{n} = 0 \\ x_{n}^{- 1} | x_{n} |^{q} & if x_{n} \neq 0 \end{matrix}$ gives $L_{x} (y) = ‖ x ‖_{q}$ , so that in fact $‖ L_{x} ‖_{(ℓ^{p})^{*}} = ‖ x ‖_{q} .$ Conversely, given a bounded linear functional ⁠ $L$ ⁠ on ⁠ $ℓ^{p}$ ⁠, the sequence defined by ⁠ $x_{n} = L (e_{n})$ ⁠ lies in ⁠ $ℓ^{q}$ ⁠. Thus the mapping ⁠ $x \mapsto L_{x}$ ⁠ gives an isometry $κ_{q} : ℓ^{q} \to (ℓ^{p})^{*} .$

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

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

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

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

$ℓ p$ spaces are increasing in $p$

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

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

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

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

Properties of $ℓ 1$ spaces

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

⁠ $K$ ⁠ is compact;
⁠ $K$ ⁠ is weakly compact;
⁠ $K$ ⁠ is bounded, closed, and equismall at infinity.

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

References

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

Bibliography

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.

[FOOTNOTEJarchow1981129–130-1] 1 2 3 Jarchow 1981, pp. 129–130.

[Debnath,_Mikusinski-2005-2] Debnath, Lokenath; Mikusinski, Piotr (2005). Hilbert Spaces with Applications. Elsevier. pp. 120–121. ISBN 978-0-12-2084386.

[FOOTNOTETrèves2006451–458-3] 1 2 Trèves 2006, pp. 451–458.

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point Sobczyk's theorem
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $b a (Σ)$ c space Banach coordinate BK Besov $B_{p, q}^{s} (ℝ)$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{λ, p} (Ω)$ ℓ^p $ℓ^{\infty}$ L^p $L^{\infty}$ weighted Schwartz $S (ℝ^{n})$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W (X, L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

v t e Fréchet spaces
Basic concepts	Fréchet space Completeness Locally convex space Metrizability Topological vector space
Main results	Anderson–Kadec theorem Eidelheit theorem Surjection of Fréchet spaces Open mapping (Banach–Schauder) Closed graph theorem
Maps	Bilinear operator form Linear map Almost open Continuous Closed Discontinuous Functional Linear Sesquilinear Seminorm Sublinear function Transpose
Properties	Barrelled Bornological Ultrabornological F-space LF-space Mackey Complete Metrizable
Types of Fréchet spaces	Banach Hilbert Tame Fréchet Graded Fréchet Fréchet lattice
Examples	Sequence space $ℝ^{ℕ}$ Space of Entire functions on $ℂ$ Fréchet coordinate (FK) space $C^{k} (U),$ and $C^{k} (K),$ and other Spaces of test functions and distributions
Advanced topics	Differentiation in Fréchet spaces LF-space
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Sequence space

Definition

Space of all sequences

ℓp spaces

c, c0 and c00

Space of all finite sequences

Other sequence spaces

Properties of ℓp spaces and the space c0

ℓp spaces are increasing in p

ℓ2 is isomorphic to all separable, infinite dimensional Hilbert spaces

Properties of ℓ1 spaces

See also