Sequence space explained

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 spaces, consisting of the -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 c₀, 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

x _{\bull} = \left(x _{n\right)} _{n} in a set X is just an X -valued map x _{\bull} : \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\left(x _{n\right)} _{n} + \left(y _{n\right)} _{n} = \left(x _{n} + y _{n\right)} _{n} , and componentwise scalar multiplication\alpha\left(x _{n\right)} _{n} = \left(\alpha x _{n\right)} _{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:

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.

spaces

See also: L-infinity.

For

0 < p < inf ty, \ell ^{p} is the subspace of K ^{\N} consisting of all sequences x _{\bull} = \left(x _{n\right)} _{n} satisfying

If

p \geq 1 , then the real-valued function \| ⋅ \| _{p} on \ell ^{p} defined by$$\backslash |x\backslash |\_p\; ~=~\; \backslash left(\backslash sum\_n|x\_n|^p\backslash right)^\; \backslash qquad\; \backslash text\; x\; \backslash in\; \backslash ell^p$$defines a norm on \ell ^{p.} In fact, \ell ^{p} is a complete metric space with respect to this norm, and therefore is a Banach space.

If

p = 2 then \ell ^{2} is also a Hilbert space when endowed with its canonical inner product, called the , defined for all x _{\bull,} y _{\bull} \in \ell ^{p} by $$\backslash langle\; x\_\backslash bull,\; y\_\backslash bull\; \backslash rangle\; ~=~\; \backslash sum\_n\; \backslash overline\; y\_n.$$The canonical norm induced by this inner product is the usual \ell ^{2} -norm, meaning that \| x \| _{2} = \sqrt{\langle x , x \rangle} for all x \in \ell ^{p.}

If

p = inf ty, then \ell ^{infty} is defined to be the space of all bounded sequences endowed with the norm$$\backslash |x\backslash |\_\backslash infty\; ~=~\; \backslash sup\_n\; |x\_n|,$$\ell ^{infty} is also a Banach space.

If

0 < p < 1 , then \ell ^{p} does not carry a norm, but rather a metric defined by$$d(x,y)\; ~=~\; \backslash sum\_n\; \backslash left|x\_n\; -\; y\_n\backslash right|^p.\backslash ,$$

c, c₀ and c₀₀

See also: c space.

A is any sequence

x _{\bull} \in K ^{\N} such that \lim _{n} x _{n} exists. The set of all convergent sequences is a vector subspace of K ^{\N} called the . Since every convergent sequence is bounded, c is a linear subspace of \ell ^{infty} . Moreover, this sequence space is a closed subspace of \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 is called a and is said to . 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 and is called the or the .

The, 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 \left(x _{nk} \right) _{k} where x _{nk} = 1 /k for the first n entries (for k = 1 , \ldots, n ) and is zero everywhere else (that is, \left(x _{nk} \right) _{k} = \left( 1 , 1 / 2 , \ldots, 1 /(n -1 ), 1 /n, 0 , 0 , \ldots\right) ) is a Cauchy sequence but it does not converge to a sequence in c _{00} .

Space of all finite sequences

Let

K ^{infty} = \left\{\left(x _{1,} x ^{\N} _{2 ,\ldots\right)\in K} : allbutfinitelymany x _{iequal0\right\}} ,

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 let

K ^{n} denote the usual Euclidean space endowed with the Euclidean topology and let \operatorname{In} _{K ^{n}} : K ^{n} \to K ^{infty} denote the canonical inclusion \operatorname{In} _{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\operatorname{Im} \left( \operatorname{In} _{K ^{n}} \right) = \left\{ \left(x _{1,} \ldots, x _{n,} 0 , 0 , \ldots \right) : x _{1,} \ldots, x _{n} \in K \right\} = K ^{n} x \left\{ ( 0 , 0 , \ldots) \right\} and consequently,K ^{infty} = c up _{n} \operatorname{Im} \left( \operatorname{In} _{K ^{n}} \right).

This family of inclusions gives

K ^{infty} a final topology \tau ^{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 Fréchet–Urysohn. The topology \tau ^{infty} is also strictly finer than the subspace topology induced on K ^{infty} by K ^{\N} .

Convergence in

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

Often, each image

\operatorname{Im} \left( \operatorname{In} _{K ^{n}} \right) is identified with the corresponding K ^{n} ; explicitly, the elements \left( x _{1,} \ldots, x _{n} \right) \in K ^{n} and \left( x _{1,} \ldots, x _{n,} 0 , 0 , 0 , \ldots \right) are identified. This is facilitated by the fact that the subspace topology on \operatorname{Im} \left( \operatorname{In} _{K ^{n}} \right) , the quotient topology from the map \operatorname{In} _{K ^{n}} , and the Euclidean topology on K ^{n} all coincide. With this identification, \left( \left( K ^{infty} , \tau ^{infty} \right), \left(\operatorname{In} _{K ^{n}} \right) _{n} \right) is the direct limit of the directed system \left( \left( K ^{n\right)} _{n \in \N} , \left(\operatorname{In} _{K ^{m\toK} ^{n}} \right) _{m} ,\N \right), where every inclusion adds trailing zeros: \operatorname{In} _{K ^{m\toK} ^{n}} \left(x _{1,} \ldots, x _{m\right)} = \left(x _{1,} \ldots, x _{m,} 0 , \ldots, 0 \right) . This shows \left( 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 for which\sup _{n} \left\vert \sum ^{n} _{i =0} x _{i} \right\vert < inf ty.

This space, when equipped with the norm

\|x\| _{bs} = \sup _{n} \left\vert \sum ^{n} _{i =0} x _{i} \right\vert,

is a Banach space isometrically isomorphic to

\ell ^{infty} , via the linear mapping(x _{n)} _{n} \mapsto \left(\sum ^{n} _{i =0} x _{i\right)} _{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.

Properties of ℓ^p spaces and the space c₀

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

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

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

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

If 1&thinsp;<&thinsp;p&thinsp;<&thinsp;∞, 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 the functional$$L\_x(y)\; =\; \backslash sum\_n\; x\_n\; y\_n$$for y in . Hölder's inequality implies that L_x is a bounded linear functional on, and in fact$$|L\_x(y)|\; \backslash le\; \backslash |x\backslash |\_q\backslash ,\backslash |y\backslash |\_p$$so that the operator norm satisfies

\|L _{x\|} ^{(\ell ^{p)} ^{*}} \stackrel{\rm{def}}{ = }\sup _{y\in\ell ^{p,} y\not =0} (|L _{x(y)|})/(\|y\| _{p}) \le \|x\| _{q.} In fact, taking y to be the element of withy _{n} = \begin{cases} 0 & if x _{n=0\\ x} ^{-1} ^{n} |x ^{q} _{n|} & if ~ x _{n} ≠ 0 \end{cases} gives L_x(y) = ||x||_q, so that in fact\|L _{x\|} ^{(\ell ^{p)} ^{*}} = \|x\| _{q.} Conversely, given a bounded linear functional L on, the sequence defined by lies in ℓ^q. Thus the mapping x\mapsto L _{x} gives an isometry$$\backslash kappa\_q\; :\; \backslash ell^q\; \backslash to\; (\backslash ell^p)^*.$$

The map

\ell ^{p)} _{q}(\ell} ^{*)} ^{q} ^{-1} }(\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, where e_i is the sequence which is zero but for a 1 in the i^&thinsp;th entry.

The space ℓ¹ has the Schur property: In ℓ¹, any sequence that is weakly convergent is also strongly convergent . 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 . That is, for every separable Banach space X, there exists a quotient map

Q:\ell ^{1} \to X , so that X is isomorphic to \ell ^{1} / \ker Q . In general, ker Q is not complemented in ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that \ell ^{1} = Y ⊕ \ker Q . In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take X = \ell ^{p} ; since there are uncountably many such Xs, and since no ℓ^p is isomorphic to any other, there are thus uncountably many ker Qs).

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

ℓ^p spaces are increasing in p

For

p\in[ 1 , inf ty] , the spaces \ell ^{p} are increasing in p , with the inclusion operator being continuous: for 1 \le p<q\le inf ty , one has \|x\| _{q\le\|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 style\sum |x ^{q} _{i|} \le 1 for q>p . But if \|x\| _{p} = 1 , then |x _{i|\le} 1 for all i , and then style\sum |x ^{q} _{i|} \le style\sum |x ^{p} _{i|} = 1 .

ℓ² 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

\aleph _{0} ).^[1] The following two items are related:

• If H is infinite dimensional, then it is isomorphic to ℓ²
• If, then H is isomorphic to

\Complex ^{N}

Properties of ℓ¹ spaces

A sequence of elements in ℓ¹ converges in the space of complex sequences ℓ¹ if and only if it converges weakly in this space. If K is a subset of this space, then the following are equivalent:

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

\varepsilon > 0 , there exists a natural number n _{\varepsilon} \geq 0 such that for all s = \left( s _{n} \right) ^{inf ty} _{n =1} \in K .

See also

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

Bibliography

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Notes and References

1. Book: Debnath . Lokenath . Mikusinski . Piotr . Hilbert Spaces with Applications . Elsevier . 978-0-12-2084386 . 120-121 . 2005.