Sequence space#ℓp spaces

A sequence $x\_\{\backslash bull\}\; =\; \backslash left(x\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}$ in a set $X$ is just an $X$-valued map $x\_\{\backslash bull\}\; :\; \backslash N\; \backslash to\; X$ whose value at $n\; \backslash in\; \backslash N$ is denoted by $x\_n$ instead of the usual parentheses notation $x(n).$

Let $\backslash mathbb\{K\}$ denote the field either of real or complex numbers. The set $\backslash mathbb\{K\}^\{\backslash N\}$ of all sequences of elements of $\backslash mathbb\{K\}$ is a vector space for componentwise addition

:$\backslash left(x\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}\; +\; \backslash left(y\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}\; =\; \backslash left(x\_n\; +\; y\_n\backslash right)\_\{n\; \backslash in\; \backslash N\},$

and componentwise scalar multiplication

:$\backslash alpha\backslash left(x\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}\; =\; \backslash left(\backslash alpha\; x\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}.$

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

As a topological space, $\backslash mathbb\{K\}^\{\backslash N\}$ is naturally endowed with the product topology. Under this topology, $\backslash mathbb\{K\}^\{\backslash N\}$ 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 $\backslash mathbb\{K\}^\{\backslash N\}$ (and thus the product topology cannot be defined by any norm).{{sfn|Jarchow|1981|pp=129-130}} Among Fréchet spaces, $\backslash mathbb\{K\}^\{\backslash N\}$ is minimal in having no continuous norms:

{{Math theorem

| name = Theorem{{sfn|Jarchow|1981|pp=129-130}}

| math_statement = Let $X$ be a Fréchet space over $\backslash mathbb\{K\}.$

Then the following are equivalent:

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

}}But the product topology is also unavoidable: $\backslash mathbb\{K\}^\{\backslash N\}$ does not admit a strictly coarser Hausdorff, locally convex topology.{{sfn|Jarchow|1981|pp=129-130}} 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.

{{See also|Lp space{{!}}L^p space|L-infinity}}

For $\backslash ell^p$ is the subspace of $\backslash mathbb\{K\}^\{\backslash N\}$ consisting of all sequences $x\_\{\backslash bull\}\; =\; \backslash left(x\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}$ satisfying

If $p\; \backslash geq\; 1,$ then the real-valued function $\backslash |\backslash cdot\backslash |\_p$ on $\backslash ell^p$ defined by

$$\backslash |x\backslash |\_p\; ~=~\; \backslash left(\backslash sum\_n|x\_n|^p\backslash right)^\{1/p\}\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; \backslash ell^p$$

defines a norm on $\backslash ell^p.$ In fact, $\backslash ell^p$ is a complete metric space with respect to this norm, and therefore is a Banach space.

If $p\; =\; 2$ then $\backslash ell^2$ is also a Hilbert space when endowed with its canonical inner product, called the {{visible anchor|Euclidean inner product}}, defined for all $x\_\backslash bull,\; y\_\backslash bull\; \backslash in\; \backslash ell^p$ by

$$\backslash langle\; x\_\backslash bull,\; y\_\backslash bull\; \backslash rangle\; ~=~\; \backslash sum\_n\; \backslash overline\{x\_n\}\; y\_n.$$

The canonical norm induced by this inner product is the usual $\backslash ell^2$-norm, meaning that $\backslash |\backslash mathbf\{x\}\backslash |\_2\; =\; \backslash sqrt\{\backslash langle\; \backslash mathbf\{x\},\; \backslash mathbf\{x\}\; \backslash rangle\}$ for all $\backslash mathbf\{x\}\; \backslash in\; \backslash ell^p.$

If $p\; =\; \backslash infty,$ then $\backslash ell^\{\backslash infty\}$ is defined to be the space of all bounded sequences endowed with the norm

$$\backslash |x\backslash |\_\backslash infty\; ~=~\; \backslash sup\_n\; |x\_n|,$$

$\backslash ell^\{\backslash infty\}$ is also a Banach space.

If then $\backslash ell^p$ does not carry a norm, but rather a metric defined by

$$d(x,y)\; ~=~\; \backslash sum\_n\; \backslash left|x\_n\; -\; y\_n\backslash right|^p.\backslash ,$$

{{See also|c space}}

A {{em|convergent sequence}} is any sequence $x\_\{\backslash bull\}\; \backslash in\; \backslash mathbb\{K\}^\{\backslash N\}$ such that $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; x\_n$ exists.

The set {{visible anchor|c|text=$c$}} of all convergent sequences is a vector subspace of $\backslash mathbb\{K\}^\{\backslash N\}$ called the c space. Since every convergent sequence is bounded, $c$ is a linear subspace of $\backslash ell^\{\backslash infty\}.$ Moreover, this sequence space is a closed subspace of $\backslash ell^\{\backslash infty\}$ 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 {{em|null sequence}} and is said to {{em|{{visible anchor|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 {{visible anchor|c0|text=$c\_0$}} and is called the {{em|{{visible anchor|space of null sequences}}}} or the {{em|{{visible anchor|space of vanishing sequences}}}}.

The {{em|{{visible anchor|space of eventually zero sequences}}}}, {{visible anchor|c00|text=$c\_\{00\},$}} 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 $\backslash left(x\_\{nk\}\backslash right)\_\{k\; \backslash in\; \backslash N\}$ where $x\_\{nk\}\; =\; 1/k$ for the first $n$ entries (for $k\; =\; 1,\; \backslash ldots,\; n$) and is zero everywhere else (that is, $\backslash left(x\_\{nk\}\backslash right)\_\{k\; \backslash in\; \backslash N\}\; =\; \backslash left(1,\; 1/2,\; \backslash ldots,\; 1/(n-1),\; 1/n,\; 0,\; 0,\; \backslash ldots\backslash right)$) is a Cauchy sequence but it does not converge to a sequence in $c\_\{00\}.$

{{anchor|Space of finite sequences}}

Let

:$\backslash mathbb\{K\}^\{\backslash infty\}=\backslash left\backslash \{\backslash left(x\_1,\; x\_2,\backslash ldots\backslash right)\backslash in\backslash mathbb\{K\}^\{\backslash N\}:\backslash text\{all\; but\; finitely\; many\; \}x\_i\backslash text\{\; equal\; \}0\backslash right\backslash \}$

,

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

For every natural number {{nowrap|$n\; \backslash in\; \backslash N$,}} let $\backslash mathbb\{K\}^n$ denote the usual Euclidean space endowed with the Euclidean topology and let $\backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\; :\; \backslash mathbb\{K\}^n\; \backslash to\; \backslash mathbb\{K\}^\{\backslash infty\}$ denote the canonical inclusion

:$\backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; \backslash ldots\; \backslash right)$.

The image of each inclusion is

:$\backslash operatorname\{Im\}\; \backslash left(\; \backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\; \backslash right)$

= \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,

:$\backslash mathbb\{K\}^\{\backslash infty\}\; =\; \backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; \backslash operatorname\{Im\}\; \backslash left(\; \backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\; \backslash right).$

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

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

Often, each image $\backslash operatorname\{Im\}\; \backslash left(\; \backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\; \backslash right)$ is identified with the corresponding $\backslash mathbb\{K\}^n$; explicitly, the elements $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n\; \backslash right)\; \backslash in\; \backslash mathbb\{K\}^n$ and $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; 0,\; \backslash ldots\; \backslash right)$ are identified. This is facilitated by the fact that the subspace topology on $\backslash operatorname\{Im\}\; \backslash left(\; \backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\; \backslash right)$, the quotient topology from the map $\backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}$, and the Euclidean topology on $\backslash mathbb\{K\}^n$ all coincide. With this identification, $\backslash left(\; \backslash left(\backslash mathbb\{K\}^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right),\; \backslash left(\backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^n\}\backslash right)\_\{n\; \backslash in\; \backslash N\}\backslash right)$ is the direct limit of the directed system $\backslash left(\; \backslash left(\backslash mathbb\{K\}^n\backslash right)\_\{n\; \backslash in\; \backslash N\},\; \backslash left(\backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^m\backslash to\backslash mathbb\{K\}^n\}\backslash right)\_\{m\; \backslash leq\; n\backslash in\backslash N\},\backslash N\; \backslash right),$ where every inclusion adds trailing zeros:

:$\backslash operatorname\{In\}\_\{\backslash mathbb\{K\}^m\backslash to\backslash mathbb\{K\}^n\}\backslash left(x\_1,\; \backslash ldots,\; x\_m\backslash right)\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_m,\; 0,\; \backslash ldots,\; 0\; \backslash right)$.

This shows $\backslash left(\backslash mathbb\{K\}^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right)$ is an LB-space.

The space of bounded series, denote by bs, is the space of sequences $x$ for which

:

This space, when equipped with the norm

:$\backslash |x\backslash |\_\{bs\}\; =\; \backslash sup\_n\; \backslash left\backslash vert\; \backslash sum\_\{i=0\}^n\; x\_i\; \backslash right\backslash vert,$

is a Banach space isometrically isomorphic to $\backslash ell^\{\backslash infty\},$ via the linear mapping

:$(x\_n)\_\{n\; \backslash in\; \backslash N\}\; \backslash mapsto\; \backslash left(\backslash sum\_\{i=0\}^n\; x\_i\backslash right)\_\{n\; \backslash in\; \backslash N\}.$

The subspace cs 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.

{{See also|c space}}

The space ℓ² 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

:$\backslash |x+y\backslash |\_p^2\; +\; \backslash |x-y\backslash |\_p^2=\; 2\backslash |x\backslash |\_p^2\; +\; 2\backslash |y\backslash |\_p^2.$

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

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

If 1 < p < ∞, 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 {{math|ℓ^q}} the functional

$$L\_x(y)\; =\; \backslash sum\_n\; x\_n\; y\_n$$

for y in {{math|ℓ^p}}. Hölder's inequality implies that L_x is a bounded linear functional on {{math|ℓ^p}}, and in fact

$$|L\_x(y)|\; \backslash le\; \backslash |x\backslash |\_q\backslash ,\backslash |y\backslash |\_p$$

so that the operator norm satisfies

:$\backslash |L\_x\backslash |\_\{(\backslash ell^p)^*\}\; \backslash stackrel\{\backslash rm\{def\}\}\{=\}\backslash sup\_\{y\backslash in\backslash ell^p,\; y\backslash not=0\}\; \backslash frac$

In fact, taking y to be the element of {{math|ℓ^p}} with

:$y\_n\; =\; \backslash begin\{cases\}$

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, so that in fact

:$\backslash |L\_x\backslash |\_\{(\backslash ell^p)^*\}\; =\; \backslash |x\backslash |\_q.$

Conversely, given a bounded linear functional L on {{math|ℓ^p}}, the sequence defined by {{math|1=x_n = L(e_n)}} lies in ℓ^q. Thus the mapping $x\backslash mapsto\; L\_x$ gives an isometry

$$\backslash kappa\_q\; :\; \backslash ell^q\; \backslash to\; (\backslash ell^p)^*.$$

The map

:$\backslash ell^q\backslash xrightarrow\{\backslash kappa\_q\}(\backslash ell^p)^*\backslash xrightarrow\{(\backslash kappa\_q^*)^\{-1\}\}(\backslash ell^q)^\{**\}$

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 c₀ is defined as the space of all sequences converging to zero, with norm identical to ||x||_∞. It is a closed subspace of ℓ^∞, hence a Banach space. The dual of c₀ is ℓ¹; the dual of ℓ¹ is ℓ^∞. For the case of natural numbers index set, the ℓ^p and c₀ are separable, with the sole exception of ℓ^∞. The dual of ℓ^∞ is the ba space.

The spaces c₀ and ℓ^p (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis {e_i | i = 1, 2,...}, 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 {{harv|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 c₀, 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 ℓ¹, was answered in the affirmative by {{harvtxt|Banach|Mazur|1933}}. That is, for every separable Banach space X, there exists a quotient map $Q:\backslash ell^1\; \backslash to\; X$, so that X is isomorphic to $\backslash ell^1\; /\; \backslash ker\; Q$. In general, ker Q is not complemented in ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that $\backslash ell^1\; =\; Y\; \backslash oplus\; \backslash ker\; Q$. In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take $X=\backslash ell^p$; 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^p is that it is not polynomially reflexive.

For $p\backslash in[1,\backslash infty]$, the spaces $\backslash ell^p$ are increasing in $p$, with the inclusion operator being continuous: for

Let H be a separable Hilbert space. Every orthogonal set in H is at most countable (i.e. has finite dimension or $\backslash ,\backslash aleph\_0\backslash ,$).{{cite book | last1 = Debnath | first1 = Lokenath | last2 = Mikusinski | first2 = Piotr | title=Hilbert Spaces with Applications | publisher=Elsevier | isbn= 978-0-12-2084386 | pages=120-121 | year=2005}} The following two items are related:

• If H is infinite dimensional, then it is isomorphic to ℓ²
• If {{math|1=dim(H) = N}}, then H is isomorphic to $\backslash Complex^N$

A sequence of elements in ℓ¹ converges in the space of complex sequences ℓ¹ if and only if it converges weakly in this space.{{sfn | Trèves | 2006 | pp=451-458}}

If K is a subset of this space, then the following are equivalent:{{sfn | Trèves | 2006 | pp=451-458}}

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

Here K being equismall at infinity means that for every $\backslash varepsilon\; >\; 0$, there exists a natural number $n\_\{\backslash varepsilon\}\; \backslash geq\; 0$ such that for all $s\; =\; \backslash left(\; s\_n\; \backslash right)\_\{n=1\}^\{\backslash infty\}\; \backslash in\; K$.

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

{{Reflist}}

• {{citation | last1=Banach | first1=Stefan | last2=Mazur | first2=S. | title=Zur Theorie der linearen Dimension | journal=Studia Mathematica | volume=4 | year=1933 | pages=100–112| doi=10.4064/sm-4-1-100-112 }}.
• {{citation | last1=Dunford | first1=Nelson| last2=Schwartz | first2=Jacob T. | title=Linear operators, volume I | publisher=Wiley-Interscience | year=1958}}.
• {{Jarchow Locally Convex Spaces}}
• {{citation | last=Pitt | first=H.R. | doi=10.1112/jlms/s1-11.3.174 | title=A note on bilinear forms | journal=J. London Math. Soc. | volume=11 | issue=3 | year=1936 | pages=174–180}}.
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{citation | last=Schur | first=J. | title=Über lineare Transformationen in der Theorie der unendlichen Reihen | journal=Journal für die reine und angewandte Mathematik|volume=151|year=1921|pages=79–111|doi=10.1515/crll.1921.151.79}}.
• {{Trèves François Topological vector spaces, distributions and kernels}}

{{Lp spaces}}

{{Banach spaces}}

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Category:Functional analysis

Category:Sequences and series